%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1507.01732v2 [cs.DS] 8 Jul 2015 %%CreationDate: Mon Jun 24 13:48:54 2019 %%Pages: 53 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z archivetempsec.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2019.06.24:1348 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro 0 0 %! 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HPSdict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(W)-7 b(e)30 b(remark)e(that)i(neither)f(the)h(time)f (slice)f(algorithm)e(nor)j(the)h(Charter)f(algorithm)523 548 y(requires)c(prior)h(kno)n(wledge)f(of)j Fv(n)p FD(,)f(the)h(n)n (um)n(b)r(er)e(of)i(items)e(due)i(to)f(arriv)n(e.)648 653 y(A)n(t)34 b(the)g(core)e(of)i(the)g(analysis)c(of)k(the)g(Charter) e(algorithm)e(w)n(e)j(pro)n(v)n(e)f(a)h(b)r(ound)h(on)523 753 y(the)i(exp)r(ected)g(size)e(of)h(the)h(maxim)n(um)c(indep)r(enden) n(t)j(set)g(of)h(a)f(random)e(unit)j(in)n(terv)-5 b(al)523 852 y(graph.)36 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b(capacit)n(y)d Fv(d)k(\015)5 b FD(-indep)r(enden)n(t)26 b(set)h(allo)n(ws)d(no)i(more)g(than)h Fv(d)523 1351 y FD(segmen)n(ts)f(o)n(v)n(erlapping)d(at)28 b(an)n(y)e(p)r(oin)n(t.) 648 1456 y(Note)g(that)h(if)g Fv(\015)g FD(=)c(1)p Fv(=n)j FD(then)h(this)f(exp)r(ected)h(size)f(is)f(ab)r(out)i(1)p Fv(=)p FD(2.)f(This)f(is)h(in)n(tuitiv)n(ely)523 1555 y(the)k(righ)n(t)f(b)r(ound)h(as)g(eac)n(h)f(in)n(terv)-5 b(al)27 b(in)j(the)g(maxim)n(um)c(indep)r(enden)n(t)k(set)g(rules)e (out)i(on)523 1655 y(a)n(v)n(erage)25 b(one)i(other)g(in)n(terv)-5 b(al)25 b(from)h(b)r(eing)h(in)g(the)h(maxim)n(um)23 b(indep)r(enden)n(t)28 b(set.)648 1760 y(W)-7 b(e)32 b(sho)n(w)g(that)g(a)g(random)e(unit)i(in)n(terv)-5 b(al)30 b(graph)h(with)g Fv(n)i FD(v)n(ertices)d(has)h(a)h(capacit)n(y)523 1860 y Fv(d)27 b(\015)5 b FD(-indep)r(enden)n(t)25 b(subset)i(of)f(exp) r(ected)h(size)e(at)h(least)f(min)n(\()p Fv(n;)14 b(d=\015)5 b FD(\)\(1)16 b Fs(\000)f Fv(\002)r FD(\()2967 1789 y Fs(p)p 3037 1789 127 4 v 3037 1860 a FD(ln)e Fv(d=)3205 1789 y Fs(p)p 3274 1789 44 4 v 71 x Fv(d)p FD(\)\).)523 1960 y(W)-7 b(e)28 b(also)e(sho)n(w)i(that)g(when)g Fv(n)c FD(=)f Fv(d=\015)33 b FD(the)28 b(exp)r(ected)h(size)d(of)i(the)h (maxim)n(um)24 b(capacit)n(y)i Fv(d)523 2059 y(\015)5 b FD(-indep)r(enden)n(t)26 b(subset)g(is)g(no)g(more)e(than)j Fv(n)p FD(\(1)16 b Fs(\000)g Fv(\002)r FD(\(1)p Fv(=)2319 1988 y Fs(p)p 2388 1988 V 71 x Fv(d)p FD(\)\).)27 b(These)f(results)f (ma)n(y)g(b)r(e)i(of)523 2159 y(indep)r(enden)n(t)g(in)n(terest.)523 2295 y Fr(Related)c(w)m(ork:)d FD(W)-7 b(orst)20 b(case)g(comp)r (etitiv)n(e)d(analysis)g(of)j(in)n(terv)-5 b(al)18 b(sc)n(heduling)g (has)i(a)g(long)523 2394 y(history)-7 b(,)33 b(e.g.,)h([)1031 2394 y SDict begin H.S end 1031 2394 a Black FD(30)p Black 1114 2341 a SDict begin H.R end 1114 2341 a 1114 2394 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Woeginger19945) cvn H.B /ANN pdfmark end 1114 2394 a FD(,)1137 2394 y SDict begin H.S end 1137 2394 a Black FD(28)p Black 1220 2341 a SDict begin H.R end 1220 2341 a 1220 2394 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lipton:1994:OIS) cvn H.B /ANN pdfmark end 1220 2394 a FD(].)h(This)e(is)g(the)i(problem)d(of)i(c)n(ho)r (osing)e(a)i(set)h(of)f(non-o)n(v)n(erlapping)523 2494 y(in)n(terv)-5 b(als)25 b(with)i(v)-5 b(arious)25 b(target)i (functions,)g(t)n(ypically)-7 b(,)24 b(the)k(sum)f(of)g(v)-5 b(alues.)648 2599 y([)671 2599 y SDict begin H.S end 671 2599 a Black FD(19)p Black 754 2546 a SDict begin H.R end 754 2546 a 754 2599 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hajiaghayi:reuse05) cvn H.B /ANN pdfmark end 754 2599 a FD(])32 b(in)n(tro)r(duce)e(the)i(question)f(of)h(auctions)e(for)h(reusable)f (go)r(o)r(ds.)h(They)h(consider)d(a)523 2699 y(w)n(orst)d(case)h(mec)n (hanism)c(design)j(setting.)g(Their)g(main)g(goal)f(is)h(addressing)e (the)k(issue)e(of)523 2798 y(time)g(incen)n(tiv)n(e)f(compatibilit)n(y) -7 b(,)23 b(for)k(some)f(restricted)g(set)h(of)h(misrepresen)n (tations.)648 2904 y(The)e(secretary)e(problem)g(is)h(arguably)f(due)i (to)g(Johannes)f(Kepler)f(\(1571-1630\),)f(and)523 3003 y(has)34 b(a)h(great)e(man)n(y)h(v)-5 b(arian)n(ts,)32 b(a)j(surv)n(ey)e(b)n(y)i([)2043 3003 y SDict begin H.S end 2043 3003 a Black FD(15)p Black 2126 2950 a SDict begin H.R end 2126 2950 a 2126 3003 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Freeman:1983) cvn H.B /ANN pdfmark end 2126 3003 a FD(])g(con)n(tains)e(some)g(70)h(references.)g(The)523 3103 y(\\p)r(erm)n(utation")27 b(mo)r(del)h(is)i(that)g(items)f(arriv)n (e)e(in)i(some)g(random)f(order,)h(all)f Fv(n)p FD(!)i(p)r(erm)n(u-)523 3202 y(tations)f(equally)f(lik)n(ely)-7 b(.)26 b(Maximizing)g(the)31 b(probabilit)n(y)26 b(that)31 b(the)f(b)r(est)h(item)e(is)g(c)n(hosen,) 523 3302 y(when)35 b(the)g(items)e(app)r(ear)h(in)g(random)f(order,)h (only)f(comparisons)e(can)j(b)r(e)i(made,)d(and)523 3402 y(the)k(n)n(um)n(b)r(er)f(of)g(items)g(is)f(kno)n(wn)h(in)g(adv)-5 b(ance,)37 b(w)n(as)e(solv)n(ed)g(b)n(y)h([)2715 3402 y SDict begin H.S end 2715 3402 a Black FD(27)p Black 2799 3348 a SDict begin H.R end 2799 3348 a 2799 3402 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lindley:1961) cvn H.B /ANN pdfmark end 2799 3402 a FD(])g(and)h(b)n(y)g([)3177 3402 y SDict begin H.S end 3177 3402 a Black FD(12)p Black 3260 3348 a SDict begin H.R end 3260 3348 a 3260 3402 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Dynkin:1963) cvn H.B /ANN pdfmark end 3260 3402 a FD(].)g(A)523 3501 y(great)28 b(man)n(y)g(other)i(v)-5 b(arian)n(ts)27 b(are)i(describ)r(ed)f(in)h (\([)2165 3501 y SDict begin H.S end 2165 3501 a Black FD(15)p Black 2248 3448 a SDict begin H.R end 2248 3448 a 2248 3501 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Freeman:1983) cvn H.B /ANN pdfmark end 2248 3501 a FD(,)2271 3501 y SDict begin H.S end 2271 3501 a Black FD(11)p Black 2354 3448 a SDict begin H.R end 2354 3448 a 2354 3501 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Dinitz:2013) cvn H.B /ANN pdfmark end 2354 3501 a FD(]\),)h(di\013ering)e(in)h(the)h(n)n (um)n(b)r(er)e(of)523 3601 y(items)22 b(to)h(b)r(e)h(c)n(hosen,)f(the)h (target)e(function)i(to)f(b)r(e)h(maximized,)19 b(taking)j(discoun)n (ting)f(in)n(to)523 3701 y(accoun)n(t,)27 b(etc.)648 3806 y(An)37 b(alternativ)n(e)c(to)k(the)g(random)e(p)r(erm)n(utation)f (mo)r(del)h(is)h(the)h(sto)r(c)n(hastic)e(arriv)-5 b(al)523 3905 y(mo)r(del,)36 b(in)n(tro)r(duced)h(b)n(y)g(Karlin)e([)1646 3905 y SDict begin H.S end 1646 3905 a Black FD(21)p Black 1729 3852 a SDict begin H.R end 1729 3852 a 1729 3905 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Karlin:1962) cvn H.B /ANN pdfmark end 1729 3905 a FD(])j(in)f(a)g(\\full)f(information")e (\(kno)n(wn)j(distribution)523 4005 y(on)c(v)-5 b(alues\))32 b(setting.)g(Bruss)g([)1489 4005 y SDict begin H.S end 1489 4005 a Black FD(7)p Black 1531 3952 a SDict begin H.R end 1531 3952 a 1531 4005 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.bruss1984) cvn H.B /ANN pdfmark end 1531 4005 a FD(])h(subsequen)n(tly)f(studied)g(the)i(sto)r(c)n(hastic)d (arriv)-5 b(al)30 b(mo)r(del)523 4105 y(in)f(a)f(no-information)d(mo)r (del)j(\(nothing)g(is)g(kno)n(wn)h(ab)r(out)g(the)g(distribution)e(of)i (v)-5 b(alues\).)523 4204 y(Recen)n(tly)e(,)25 b([)903 4204 y SDict begin H.S end 903 4204 a Black FD(13)p Black 986 4151 a SDict begin H.R end 986 4151 a 986 4204 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.FNS11) cvn H.B /ANN pdfmark end 986 4204 a FD(])h(made)f(use)h(of)g(the)h(sto)r(c)n (hastic)d(arriv)-5 b(al)22 b(mo)r(del)j(as)g(a)h(to)r(ol)e(for)i(the)g (analysis)523 4304 y(of)i(algorithms)23 b(in)k(the)h(p)r(erm)n(utation) d(mo)r(del.)648 4409 y(Muc)n(h)f(of)g(the)h(recen)n(t)f(in)n(terest)f (in)h(the)h(secretary)d(problem)g(is)i(due)g(to)h(it's)e(connection)523 4509 y(to)28 b(incen)n(tiv)n(e)d(compatible)f(auctions)i(and)h(p)r (osted)h(prices)e([)2411 4509 y SDict begin H.S end 2411 4509 a Black FD(18)p Black 2494 4455 a SDict begin H.R end 2494 4455 a 2494 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Hajiaghayi:2004) cvn H.B /ANN pdfmark end 2494 4509 a FD(,)2517 4509 y SDict begin H.S end 2517 4509 a Black FD(24)p Black 2600 4455 a SDict begin H.R end 2600 4455 a 2600 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kleinberg) cvn H.B /ANN pdfmark end 2600 4509 a FD(,)2623 4509 y SDict begin H.S end 2623 4509 a Black FD(2)p Black 2665 4455 a SDict begin H.R end 2665 4455 a 2665 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff07-knapsack) cvn H.B /ANN pdfmark end 2665 4509 a FD(,)2688 4509 y SDict begin H.S end 2688 4509 a Black FD(3)p Black 2729 4455 a SDict begin H.R end 2729 4455 a 2729 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff08) cvn H.B /ANN pdfmark end 2729 4509 a FD(,)2752 4509 y SDict begin H.S end 2752 4509 a Black FD(1)p Black 2794 4455 a SDict begin H.R end 2794 4455 a 2794 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff:2009:SPW) cvn H.B /ANN pdfmark end 2794 4509 a FD(,)2817 4509 y SDict begin H.S end 2817 4509 a Black FD(10)p Black 2900 4455 a SDict begin H.R end 2900 4455 a 2900 4509 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.devanur2009adwords) cvn H.B /ANN pdfmark end 2900 4509 a FD(].)648 4614 y(Most)d(directly)e(relev)-5 b(an)n(t)22 b(to)i(this)f(pap)r(er)g (is)g(the)h Fv(k)s FD(-secretary)e(algorithm)d(b)n(y)24 b(R.)g(Klein-)523 4714 y(b)r(erg)29 b([)735 4714 y SDict begin H.S end 735 4714 a Black FD(24)p Black 818 4660 a SDict begin H.R end 818 4660 a 818 4714 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kleinberg) cvn H.B /ANN pdfmark end 818 4714 a FD(].)g(Constrained)f(to)h(pic)n(king)e(no)i(more)e (than)i Fv(k)k FD(secretaries,)26 b(the)k(total)e(v)-5 b(alue)28 b(of)523 4813 y(the)e(secretaries)d(pic)n(k)n(ed)i(b)n(y)g (this)g(algorithm)d(is)j(at)h(least)e(a)i(\(1)15 b Fs(\000)2619 4781 y FC(5)p 2590 4795 92 4 v 2590 4803 a Fq(p)p 2645 4803 37 3 v 50 x Ft(k)2692 4813 y FD(\))26 b(of)g(the)g(v)-5 b(alue)25 b(of)h(the)523 4924 y(b)r(est)i Fv(k)j FD(secretaries.)p Black 1943 5173 a(3)p Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Babaio\013)26 b Fw(et)i(al.)f FD([)1213 448 y SDict begin H.S end 1213 448 a Black FD(2)p Black 1255 395 a SDict begin H.R end 1255 395 a 1255 448 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff07-knapsack) cvn H.B /ANN pdfmark end 1255 448 a FD(])h(in)n(tro)r(duced)f(the)h Fp(knapsack)k(se)l(cr)l(etary)e(pr)l(oblem)g FD(in)d(whic)n(h)g(ev)n (ery)523 548 y(secretary)38 b(has)g(some)g(w)n(eigh)n(t)g(and)h(a)g(v) -5 b(alue,)38 b(and)h(one)g(seeks)g(to)g(maximize)c(the)40 b(sum)523 648 y(of)c(v)-5 b(alues)35 b(sub)5 b(ject)36 b(to)h(a)e(upp)r(er)i(b)r(ound)f(on)g(the)h(total)e(w)n(eigh)n(t.)f (They)j(giv)n(e)d(a)i(1)p Fv(=)p FD(\(10)p Fv(e)p FD(\))523 747 y(comp)r(etitiv)n(e)26 b(algorithm)g(for)i(this)h(problem.)e (\(Note)i(that)h(if)f(w)n(eigh)n(ts)e(are)h(one)h(then)h(this)523 847 y(b)r(ecomes)f(the)i Fv(k)s FD(-secretary)e(problem\).)f(The)j (Matroid)d(secretary)h(problem,)f(in)n(tro)r(duced)523 946 y(b)n(y)20 b(Babaio\013)e(et)j(al.)d([)1172 946 y SDict begin H.S end 1172 946 a Black FD(4)p Black 1214 893 a SDict begin H.R end 1214 893 a 1214 946 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.DBLP:conf/soda/BabaioffIK07) cvn H.B /ANN pdfmark end 1214 946 a FD(],)i(constrains)e(the)j(set)f(of)g(secretaries)d (pic)n(k)n(ed)i(to)h(b)r(e)g(an)g(indep)r(enden)n(t)523 1046 y(set)32 b(in)f(some)f(underlying)f(Matroid.)h(Subsequen)n(t)h (results)f(for)i(arbitrary)c(Matrio)r(ds)i(are)523 1146 y(giv)n(en)c(in)h([)860 1146 y SDict begin H.S end 860 1146 a Black FD(8)p Black 901 1092 a SDict begin H.R end 901 1092 a 901 1146 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Chakraborty:2012) cvn H.B /ANN pdfmark end 901 1146 a FD(,)924 1146 y SDict begin H.S end 924 1146 a Black FD(26)p Black 1007 1092 a SDict begin H.R end 1007 1092 a 1007 1146 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lachish14) cvn H.B /ANN pdfmark end 1007 1146 a FD(,)1030 1146 y SDict begin H.S end 1030 1146 a Black FD(14)p Black 1113 1092 a SDict begin H.R end 1113 1092 a 1113 1146 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.FeldmanSZ15) cvn H.B /ANN pdfmark end 1113 1146 a FD(].)648 1253 y(Another)j(generalization)c(of)31 b(the)g(secretary)e(problem)f(is)i(the)h(online)d(maxim)n(um)f(bi-)523 1352 y(partite)c(matc)n(hing)e(problem.)g(See)j([)1659 1352 y SDict begin H.S end 1659 1352 a Black FD(25)p Black 1742 1299 a SDict begin H.R end 1742 1299 a 1742 1352 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.KorulaP09) cvn H.B /ANN pdfmark end 1742 1352 a FD(,)1765 1352 y SDict begin H.S end 1765 1352 a Black FD(22)p Black 1848 1299 a SDict begin H.R end 1848 1299 a 1848 1352 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.KesselheimRTV13) cvn H.B /ANN pdfmark end 1848 1352 a FD(].)h(Secretary)d(mo)r(dels)g(with)h(full)g (information)d(or)523 1452 y(partial)g(information)f(\(priors)i(on)h(v) -5 b(alues\))22 b(app)r(ear)f(in)h([)2264 1452 y SDict begin H.S end 2264 1452 a Black FD(5)p Black 2306 1398 a SDict begin H.R end 2306 1398 a 2306 1452 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Badanidiyuru:2012:LBP) cvn H.B /ANN pdfmark end 2306 1452 a FD(])h(and)f([)2531 1452 y SDict begin H.S end 2531 1452 a Black FD(29)p Black 2614 1398 a SDict begin H.R end 2614 1398 a 2614 1452 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Singer:2013:PMC) cvn H.B /ANN pdfmark end 2614 1452 a FD(].)h(This)f(w)n(as)g(in)g(the)h(con-)523 1552 y(text)j(of)f(submo)r(dular)d(pro)r(curemen)n(t)i(auctions)f(\([) 2101 1552 y SDict begin H.S end 2101 1552 a Black FD(5)p Black 2143 1498 a SDict begin H.R end 2143 1498 a 2143 1552 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Badanidiyuru:2012:LBP) cvn H.B /ANN pdfmark end 2143 1552 a FD(]\))j(and)f(budget)g(feasible) e(pro)r(curemen)n(t)523 1651 y(\([)578 1651 y SDict begin H.S end 578 1651 a Black FD(29)p Black 661 1598 a SDict begin H.R end 661 1598 a 661 1651 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Singer:2013:PMC) cvn H.B /ANN pdfmark end 661 1651 a FD(]\).)28 b(Other)g(pap)r(ers)e(considering)f(a)i(sto)r (c)n(hastic)f(setting)g(include)g([)2742 1651 y SDict begin H.S end 2742 1651 a Black FD(23)p Black 2825 1598 a SDict begin H.R end 2825 1598 a 2825 1651 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.KesselheimTRV14) cvn H.B /ANN pdfmark end 2825 1651 a FD(,)2848 1651 y SDict begin H.S end 2848 1651 a Black FD(17)p Black 2931 1598 a SDict begin H.R end 2931 1598 a 2931 1651 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.HKSV14) cvn H.B /ANN pdfmark end 2931 1651 a FD(].)648 1758 y(In)21 b(our)g(analysis,)d(w)n(e)j(giv)n(e)f(a)h (detailed)e(and)j(quite)e(tec)n(hnical)g(lo)n(w)n(er)e(b)r(ound)k(on)f (the)h(size)523 1858 y(of)27 b(the)g(maxim)n(um)c(indep)r(enden)n(t)k (set)g(in)g(a)f(random)f(unit)i(in)n(terv)-5 b(al)24 b(graph)i(\(pro)r(duced)h(b)n(y)523 1958 y(the)g(greedy)f(algorithm\).) d(Indep)r(enden)n(t)28 b(sets)f(in)f(other)h(random)e(in)n(terv)-5 b(al)24 b(graph)i(mo)r(dels)523 2057 y(w)n(ere)h(previously)d(studied)j (in)g([)1523 2057 y SDict begin H.S end 1523 2057 a Black FD(20)p Black 1606 2004 a SDict begin H.R end 1606 2004 a 1606 2057 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.winkler1990) cvn H.B /ANN pdfmark end 1606 2057 a FD(,)1629 2057 y SDict begin H.S end 1629 2057 a Black FD(9)p Black 1671 2004 a SDict begin H.R end 1671 2004 a 1671 2057 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.winkler95) cvn H.B /ANN pdfmark end 1671 2057 a FD(,)1694 2057 y SDict begin H.S end 1694 2057 a Black FD(6)p Black 1735 2004 a SDict begin H.R end 1735 2004 a 1735 2057 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.CPC:91295) cvn H.B /ANN pdfmark end 1735 2057 a FD(].)523 2257 y SDict begin H.S end 523 2257 a 523 2257 a SDict begin 12 H.A end 523 2257 a 523 2257 a SDict begin [/View [/XYZ H.V]/Dest (section.1.2) cvn /DEST pdfmark end 523 2257 a 103 x Fx(2)112 b(F)-9 b(ormal)38 b(Statemen)m(t)g(of)g(Problems)g(Considered)523 2596 y FD(Eac)n(h)c(item)g Fv(x)h FD(has)g(a)g(v)-5 b(alue)34 b Fv(v)s FD(\()p Fv(x)p FD(\),)j(w)n(e)d(assume)g(that)h(for)g(all)e Fv(x)j Fs(6)p FD(=)f Fv(y)s FD(,)g Fv(v)s FD(\()p Fv(x)p FD(\))i Fs(6)p FD(=)f Fv(v)s FD(\()p Fv(y)s FD(\))f(b)n(y)523 2696 y(consisten)n(t)c(tie)i(breaking,)d(and)j(w)n(e)f(sa)n(y)g(that)h Fv(x)g(>)e(y)36 b FD(i\013)c Fv(v)s FD(\()p Fv(x)p FD(\))h Fv(>)f(v)s FD(\()p Fv(y)s FD(\).)h(Giv)n(en)f(a)g(set)h(of)523 2795 y(items)26 b Fv(X)7 b FD(,)27 b(de\014ne)h Fv(v)s FD(\()p Fv(X)7 b FD(\))23 b(=)1405 2733 y Fu(P)1492 2820 y Ft(x)p Fq(2)p Ft(X)1652 2795 y Fv(v)s FD(\()p Fv(x)p FD(\))28 b(and)g Fv(T)2045 2807 y Ft(k)2085 2795 y FD(\()p Fv(X)7 b FD(\))23 b(=)g(max)2490 2810 y Ft(T)9 b Fq(\022)p Ft(X)q(;)p Fq(j)p Ft(T)g Fq(j\024)p Ft(k)2860 2795 y Fv(v)s FD(\()p Fv(T)j FD(\).)648 2902 y(Giv)n(en)35 b(a)i(set)g Fv(X)44 b FD(and)37 b(a)f(densit)n(y)g(distribution)e(function)j Fv(f)46 b FD(de\014ned)37 b(on)g([0)p Fv(;)14 b FD(1\),)37 b(let)523 3002 y Fv(\022)562 3014 y Ft(f)628 3002 y FD(:)23 b Fv(X)30 b Fs(7!)23 b FD([0)p Fv(;)14 b FD(1\))26 b(b)r(e)h(a)g (random)d(mapping)h(where)h Fv(\022)2184 3014 y Ft(f)2227 3002 y FD(\()p Fv(x)p FD(\))i(is)e(dra)n(wn)g(indep)r(enden)n(tly)f (from)523 3102 y(the)f(distribution)c Fv(f)9 b FD(.)23 b(The)g(function)g Fv(\022)1732 3114 y Ft(f)1798 3102 y FD(is)g(called)d(a)j Fw(sto)r(c)n(hastic)f(arriv)-5 b(al)19 b FD(function,)k(and)g(w)n(e)523 3201 y(in)n(terpret)29 b Fv(\022)909 3213 y Ft(f)952 3201 y FD(\()p Fv(x)p FD(\),)i Fv(x)c Fs(2)h Fv(X)7 b FD(,)29 b(to)h(b)r(e)h(the)f(time)f(at)g(whic)n (h)g(item)g Fv(x)i FD(arriv)n(es.)26 b(F)-7 b(or)30 b(the)g(sp)r(ecial) 523 3301 y(case)d(in)g(whic)n(h)f Fv(f)36 b FD(is)27 b(uniform)e(w)n(e)j(refer)f(to)g Fv(\022)1960 3313 y Ft(f)2031 3301 y FD(as)g Fv(\022)r FD(.)648 3408 y(In)35 b(the)h(problems)c(w)n(e)j(consider,)f(the)h(items)f(arriv)n(e)e(in)j (increasing)d(order)i(of)h Fv(\022)3248 3420 y Ft(f)3291 3408 y FD(.)h(If)523 3507 y Fv(\022)562 3519 y Ft(f)605 3507 y FD(\()p Fv(x)p FD(\))i(=)f Fv(\022)895 3519 y Ft(f)938 3507 y FD(\()p Fv(y)s FD(\))g(the)f(relativ)n(e)d(order)i(of)h (arriv)-5 b(al)33 b(of)j Fv(x)g FD(and)g Fv(y)j FD(is)c(arbitrary)-7 b(.)34 b(An)i(online)523 3607 y(algorithm)28 b(ma)n(y)j(select)g(an)h (item)f(only)g(up)r(on)i(arriv)-5 b(al.)29 b(If)k(an)f(item)f Fv(x)i FD(w)n(as)e(selected,)g(w)n(e)523 3707 y(sa)n(y)26 b(that)i(the)g(online)e(algorithm)d Fp(holds)29 b Fv(x)g FD(for)e Fv(\015)32 b FD(time)26 b(follo)n(wing)d Fv(\022)2678 3719 y Ft(f)2722 3707 y FD(\()p Fv(x)p FD(\).)648 3814 y(An)h(online)e(algorithm)e Fv(A)25 b FD(for)e(the)i(temp)f(secretary)e (problem)g(ma)n(y)g(hold)h(at)h(most)f(one)523 3913 y(item)28 b(at)h(an)n(y)g(time)f(and)h(ma)n(y)f(select)g(at)h(most)f Fv(k)33 b FD(items)27 b(in)i(total.)f(W)-7 b(e)30 b(refer)e(to)i Fv(k)i FD(as)d(the)523 4013 y Fp(budget)h FD(of)f Fv(A)p FD(.)h(The)f(goal)e(of)j(the)g(algorithm)25 b(is)j(to)h(maximize)c(the) 30 b(exp)r(ected)g(total)e(v)-5 b(alue)523 4113 y(of)32 b(the)g(items)e(that)j(it)e(selects.)g(W)-7 b(e)32 b(denote)g(b)n(y)f Fv(A)p FD(\()p Fv(X)r(;)14 b(\022)2333 4125 y Ft(f)2377 4113 y FD(\))32 b(the)g(set)g(of)g(items)e(c)n(hosen)h(b)n(y)523 4212 y(algorithm)f Fv(A)35 b FD(on)f(items)e(in)i Fv(X)41 b FD(app)r(earing)32 b(according)f(to)j(sto)r(c)n(hastic)f(arriv)-5 b(al)30 b(function)523 4312 y Fv(\022)562 4324 y Ft(f)605 4312 y FD(.)648 4419 y(The)36 b(set)f(of)h(the)h(arriv)-5 b(al)32 b(times)i(of)i(the)g(items)e(selected)h(b)n(y)h(an)f(algorithm) d(for)k(the)523 4518 y(temp)25 b(secretary)f(problem)f(is)h(said)g(to)i (b)r(e)g Fv(\015)5 b Fp(-indep)l(endent)p FD(.)25 b(F)-7 b(ormally)g(,)22 b(a)j(set)h Fv(S)i Fs(\032)22 b FD([0)p Fv(;)14 b FD(1\))25 b(is)523 4618 y(said)h(to)i(b)r(e)g Fv(\015)5 b Fp(-indep)l(endent)35 b FD(if)27 b(for)g(all)f Fv(t)1801 4630 y FC(1)1838 4618 y Fv(;)14 b(t)1905 4630 y FC(2)1965 4618 y Fs(2)23 b Fv(S)5 b FD(,)28 b Fv(t)2180 4630 y FC(1)2240 4618 y Fs(6)p FD(=)23 b Fv(t)2358 4630 y FC(2)2423 4618 y FD(w)n(e)k(ha)n(v)n(e)f(that)i Fs(j)p Fv(t)2969 4630 y FC(1)3025 4618 y Fs(\000)18 b Fv(t)3138 4630 y FC(2)3175 4618 y Fs(j)23 b(\025)g Fv(\015)5 b FD(.)648 4725 y(Giv)n(en)29 b Fv(\015)34 b(>)28 b FD(0,)j(a)g(budget)g Fv(k)s FD(,)g(a)g(set)g Fv(X)37 b FD(of)31 b(items,)f(and)g(a)h (mapping)e Fv(\022)2890 4737 y Ft(f)2962 4725 y FD(:)g Fv(X)35 b Fs(7!)29 b FD([0)p Fv(;)14 b FD(1\))523 4825 y(w)n(e)29 b(de\014ne)g(Opt\()p Fv(X)r(;)14 b(\022)1210 4837 y Ft(f)1253 4825 y FD(\))29 b(to)g(b)r(e)g(a)g Fv(\015)5 b FD(-indep)r(enden)n(t)28 b(set)h Fv(S)5 b FD(,)29 b Fs(j)p Fv(S)5 b Fs(j)25 b(\024)g Fv(k)s FD(,)k(that)g(maximizes)c(the) 523 4924 y(sum)i(of)g(v)-5 b(alues.)p Black 1943 5173 a(4)p Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Giv)n(en)35 b(ren)n(tal)f(p)r(erio)r(d)i Fv(\015)42 b(>)37 b FD(0,)f(distribution)e Fv(f)9 b FD(,)36 b(and)g(budget)h Fv(k)s FD(,)f(the)h(comp)r(etitiv)n(e)523 548 y(ratio)26 b(of)h(an)g(online)f(algorithm)d Fv(A)28 b 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b(generally)-7 b(,)17 b(one)k(can)f(de\014ne)h(a)f(temp)g(secretary)f(problem)f(with)i (resp)r(ect)g(to)h(some)523 2141 y(arbitrary)h(predicate)i Fv(P)37 b FD(that)26 b(holds)e(on)h(the)h(set)f(of)h(items)d(held)i(b)n (y)g(an)g(online)e(algorithm)523 2241 y(at)33 b(all)e(times)g Fv(t)p FD(.)j(This)e(framew)n(ork)e(includes)h(all)g(of)j(the)f(v)-5 b(arian)n(ts)31 b(ab)r(o)n(v)n(e.)h(The)h(optimal)523 2340 y(solution)25 b(with)i(resp)r(ect)h(to)f Fv(P)40 b FD(is)26 b(also)g(w)n(ell)f(de\014ned.)523 2521 y SDict begin H.S end 523 2521 a 523 2521 a SDict begin 12 H.A end 523 2521 a 523 2521 a SDict begin [/View [/XYZ H.V]/Dest (section.1.3) cvn /DEST pdfmark end 523 2521 a 103 x Fx(3)112 b(The)38 b(time-slice)g(Algorithm.)523 2842 y FD(In)31 b(this)f(section)f(w)n(e)h(describ)r(e)g(a)g(simple)d(time)j (slicing)d(tec)n(hnique.)i(This)h(giv)n(es)e(a)j(reduc-)523 2942 y(tion)e(from)f(temp)h(secretary)f(problems,)f(with)i(arbitrary)e (kno)n(wn)i(prior)f(distribution)e(on)523 3041 y(arriv)-5 b(al)36 b(times,)h(to)i(the)g(\\usual")e(con)n(tin)n(uous)g(setting)h (where)h(secretaries)d(arriv)n(e)g(o)n(v)n(er)523 3141 y(time,)d(do)h(not)g(depart)g(if)g(hired,)f(and)h(the)g(distribution)e (on)i(arriv)-5 b(al)30 b(times)j(is)g(uniform.)523 3240 y(The)27 b(reduction)f(is)f(v)-5 b(alid)25 b(for)i(man)n(y)e(v)-5 b(arian)n(ts)25 b(of)i(the)g(temp)f(secretary)f(problem,)g(includ-)523 3340 y(ing)37 b(the)h(Matroid)e(secretary)g(problem,)f(and)j(the)g (knapsac)n(k)e(secretary)g(problem.)g(W)-7 b(e)523 3440 y(remark)30 b(that)j(although)e(the)i(Matrio)r(d)e(and)h(Knapsac)n(k)f (algorithms)d(are)j(stated)i(in)f(the)523 3539 y(random)26 b(p)r(erm)n(utation)f(mo)r(del,)h(they)h(can)h(b)r(e)g(replaced)d(with) i(analogous)e(algorithms)e(in)523 3639 y(the)28 b(con)n(tin)n(uous)e (time)g(mo)r(del)f(and)j(can)f(therefore)g(b)r(e)h(used)f(in)g(our)g (con)n(text.)648 3742 y(W)-7 b(e)23 b(demonstrate)f(this)h(tec)n (hnique)f(b)n(y)h(applying)e(it)i(to)g(the)h(classical)19 b(secretary)j(prob-)523 3842 y(lem)29 b(\(hire)g(the)h(b)r(est)h (secretary\).)e(W)-7 b(e)30 b(obtain)f(an)h(algorithm)c(whic)n(h)k(w)n (e)f(call)f Fv(S)5 b(l)r(ice)3234 3854 y Ft(\015)3306 3842 y FD(for)523 3941 y(the)36 b(temp)f(secretary)f(problem)f(with)i (arbitrary)e(prior)h(distribution)e(on)k(arriv)-5 b(al)32 b(times)523 4041 y(that)c(is)e Fv(O)r FD(\(1\))j(comp)r(etitiv)n(e.)648 4144 y(Consider)38 b(the)i(1)p Fv(=)p FD(2)p Fv(\015)k FD(time)39 b(in)n(terv)-5 b(als)37 b(\()p Fw(i.e.)j FD(slices\))e Fv(I)2395 4156 y Ft(j)2474 4144 y FD(=)44 b([2)p Fv(\015)5 b(j;)54 b FD(2)p Fv(\015)5 b FD(\()p Fv(j)31 b FD(+)26 b(1\)\),)41 b(0)i Fs(\024)523 4244 y Fv(j)i Fs(\024)40 b FD(1)p Fv(=)p FD(\(2)p Fv(\015)5 b FD(\))24 b Fs(\000)h FD(1.)38 b(W)-7 b(e)38 b(split)e(ev)n(ery)h(suc)n(h)h(in)n(terv)-5 b(al)35 b(in)n(to)i(t)n(w)n(o,)g Fv(I)2660 4214 y Ft(`)2653 4266 y(j)2732 4244 y FD(=)j([2)p Fv(\015)5 b(j;)52 b FD(2)p Fv(\015)5 b(j)29 b FD(+)c Fv(\015)5 b FD(\),)523 4353 y Fv(I)566 4323 y Ft(r)559 4374 y(j)626 4353 y FD(=)23 b([2)p Fv(\015)5 b(j)22 b FD(+)c Fv(\015)5 b(;)41 b FD(2)p Fv(\015)5 b FD(\()p Fv(j)23 b FD(+)18 b(1\)\).)1469 4353 y SDict begin H.S end 1469 4353 a -30 x FC(1)1506 4353 y SDict begin 12 H.L end 1506 4353 a 1506 4353 a SDict begin [/Subtype /Link/Dest (Hfootnote.1) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 1506 4353 a 648 4456 a FD(Initially)-7 b(,)30 b(w)n(e)k(\015ip)g(a)g(fair)f(coin)g(and)h(with)g(probabilit)n(y)c(1)p Fv(=)p FD(2)j(decide)h(to)g(pic)n(k)f(p)r(oin)n(ts)523 4556 y(only)24 b(from)g(the)i(left)f(halv)n(es)f(\()p Fv(I)1504 4526 y Ft(`)1497 4577 y(j)1536 4556 y FD('s\))i(or)e(only)g (from)g(the)i(righ)n(t)e(halv)n(es)f(\()p Fv(I)2785 4526 y Ft(r)2778 4577 y(j)2822 4556 y FD('s\).)j(In)g(eac)n(h)e(suc)n(h)523 4655 y(in)n(terv)-5 b(al)28 b(w)n(e)h(pic)n(k)g(at)h(most)f(one)h(item) e(b)n(y)i(running)f(the)i(follo)n(wing)26 b(mo)r(di\014cation)h(of)j (the)523 4755 y(con)n(tin)n(uous)c(time)g(secretary)f(algorithm.)p Black 523 4839 473 4 v 546 4893 a Fn(1)606 4924 y SDict begin H.S end 606 4924 a 606 4924 a SDict begin H.R end 606 4924 a 606 4924 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.1) cvn /DEST pdfmark end 606 4924 a FB(F)-6 b(or)26 b(simplicit)n(y)f(w)n(e)h(assume)g(that)f(1)p Fy(=)p FB(\(2)p Fy(\015)t FB(\))h(is)h(an)e(in)n(teger.)p Black Black 1943 5173 a FD(5)p Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(The)40 b(con)n(tin)n(uous)e(time)h(secretary)g (algorithm)d([)2242 448 y SDict begin H.S end 2242 448 a Black FD(13)p Black 2325 395 a SDict begin H.R end 2325 395 a 2325 448 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.FNS11) cvn H.B /ANN pdfmark end 2325 448 a FD(])41 b(observ)n(es)d (the)j(items)e(arriving)523 548 y(b)r(efore)31 b(time)e(1)p Fv(=e)p FD(,)h(sets)h(the)h(largest)c(v)-5 b(alue)30 b(of)h(an)g(observ)n(ed)f(item)f(as)i(a)g(threshold,)e(and)523 648 y(then)23 b(c)n(ho)r(oses)e(the)i(\014rst)f(item)f(\(that)j(arriv)n (es)19 b(follo)n(wing)g(time)i(1)p Fv(=e)p FD(\))g(of)i(v)-5 b(alue)21 b(greater)g(than)523 747 y(the)26 b(threshold.)e(The)i(mo)r (di\014ed)e(con)n(tin)n(uous)f(time)h(secretary)g(algorithm)d(observ)n (es)j(items)523 847 y(as)30 b(long)f(as)g(the)i(cum)n(ulativ)n(e)c (distribution)h(function)i(of)g(the)h(curren)n(t)f(time)f(is)g(less)g (than)523 946 y(1)p Fv(=e)p FD(,)d(then)i(it)e(sets)h(the)h(largest)d (v)-5 b(alue)26 b(of)h(an)g(observ)n(ed)e(item)h(as)h(a)f(threshold)g (compute)g(a)523 1046 y(threshold,)g(and)i(c)n(ho)r(ose)e(the)i(next)g (item)e(of)h(v)-5 b(alue)27 b(larger)e(than)i(the)h(threshold.)648 1146 y(It)34 b(is)f(clear)f(that)i(an)n(y)g(t)n(w)n(o)f(p)r(oin)n(ts)g (pic)n(k)n(ed)f(b)n(y)i(this)f(algorithm)d(ha)n(v)n(e)j(arriv)-5 b(al)31 b(times)523 1245 y(separated)26 b(b)n(y)i(at)f(least)f Fv(\015)5 b FD(.)523 1261 y SDict begin H.S end 523 1261 a 523 1261 a SDict begin 12 H.A end 523 1261 a 523 1261 a SDict begin [/View [/XYZ H.V]/Dest (theorem.1.3.1) cvn /DEST pdfmark end 523 1261 a Black 131 x Fr(Theorem)30 b(1.)p Black 41 w Fp(The)h(algorithm)g Fv(S)5 b(l)r(ice)1773 1404 y Ft(\015)1844 1392 y Fp(is)30 b FD(1)p Fv(=)p FD(\(2)p Fv(e)p FD(\))e Fp(c)l(omp)l(etitive.)p Black 523 1538 a(Pr)l(o)l(of.)p Black 43 w FD(The)k(analysis)d(is)i(as)h(follo)n(ws.)d (Fix)j(the)g(mapping)e(of)i(items)f(to)h(eac)n(h)g(of)g(the)g(left)523 1638 y(in)n(terv)-5 b(als)24 b Fv(I)898 1607 y Ft(`)891 1659 y(j)930 1638 y FD('s)j(and)g(to)f(eac)n(h)g(of)h(the)g(righ)n(t)f (in)n(terv)-5 b(als)24 b Fv(I)2272 1607 y Ft(r)2265 1659 y(j)2308 1638 y FD('s)j(\(lea)n(ving)d(free)j(the)g(assignmen)n(t)523 1737 y(of)f(items)e(to)i(sp)r(eci\014c)f(arriv)-5 b(al)22 b(times)i(within)g(their)h(the)h(in)n(terv)-5 b(als)23 b(they)j(are)f(assigned)f(to\).)523 1837 y(Let)i Fv(O)r(P)12 b(T)861 1807 y Ft(`)919 1837 y FD(\()p Fv(O)r(P)g(T)1142 1807 y Ft(r)1178 1837 y FD(\))27 b(b)r(e)f(the)g(sum)g(of)g(the)g (items)e(of)i(maxim)n(um)c(v)-5 b(alue)25 b(o)n(v)n(er)f(all)g(in)n (terv)-5 b(als)523 1937 y Fv(I)566 1906 y Ft(`)559 1958 y(j)627 1937 y FD(\()p Fv(I)702 1906 y Ft(r)695 1958 y(j)739 1937 y FD(\).)29 b(Let)f Fv(O)r(P)12 b(T)40 b FD(b)r(e)29 b(the)g(a)n(v)n(erage)d(optimal)f(v)-5 b(alue)27 b(conditioned)g(on)h(this)g(mapping)e(of)523 2036 y(items)g(to)h(in)n (terv)-5 b(als.)25 b(Clearly)-7 b(,)1529 2191 y SDict begin H.S end 1529 2191 a 1529 2191 a SDict begin 12 H.A end 1529 2191 a 1529 2191 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.3.2) cvn /DEST pdfmark end 1529 2191 a Fv(O)r(P)12 b(T)1720 2156 y Ft(`)1769 2191 y FD(+)18 b Fv(O)r(P)12 b(T)2043 2156 y Ft(r)2103 2191 y Fs(\025)22 b Fv(O)r(P)12 b(T)7 b(:)900 b FD(\(2\))648 2345 y(F)-7 b(or)26 b(an)n(y)h(in)n(terv)-5 b(al)25 b Fv(I)1289 2357 y Ft(j)1324 2345 y FD('s)j(\()p Fv(I)1483 2315 y Ft(`)1476 2367 y(j)1515 2345 y FD('s\))g Fv(S)5 b(l)r(ice)1818 2357 y Ft(\015)1886 2345 y FD(gain)26 b(at)h(least)f(1)p Fv(=e)h FD(o)n(v)n(er)e(the)j(top)g(v)-5 b(alue)26 b(in)h(the)523 2445 y(in)n(terv)-5 b(al)31 b(conditioned)g(on)i(the)h(ev)n(en)n(t)f (that)h Fv(S)5 b(l)r(ice)2150 2457 y Ft(\015)2224 2445 y FD(do)r(esn't)34 b(ignore)d(this)i(in)n(terv)-5 b(al,)31 b(this)523 2545 y(happ)r(ens)24 b(with)g(probabilit)n(y)c(1)p Fv(=)p FD(2.)j(Therefore)g(the)i(exp)r(ected)f(sum)f(of)h(v)-5 b(alues)23 b(ac)n(hiev)n(ed)f(b)n(y)523 2644 y Fv(S)5 b(l)r(ice)710 2656 y Ft(\015)779 2644 y FD(is)26 b(at)i(least)1480 2786 y SDict begin H.S end 1480 2786 a 1480 2786 a SDict begin 12 H.A end 1480 2786 a 1480 2786 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.3.3) cvn /DEST pdfmark end 1480 2786 a 1490 2730 a FD(1)p 1490 2767 42 4 v 1490 2843 a(2)1560 2786 y Fs(\001)1612 2730 y FD(1)p 1612 2767 V 1613 2843 a Fv(e)1663 2786 y(O)r(P)12 b(T)1854 2752 y Ft(`)1904 2786 y FD(+)1997 2730 y(1)p 1997 2767 V 1997 2843 a(2)2067 2786 y Fs(\001)2118 2730 y FD(1)p 2118 2767 V 2119 2843 a Fv(e)2170 2786 y(O)r(P)g(T)2361 2752 y Ft(r)2425 2786 y Fv(:)851 b FD(\(3\))523 2947 y(Substitution)27 b(\()1031 2947 y SDict begin H.S end 1031 2947 a Black FD(2)p Black 1072 2893 a SDict begin H.R end 1072 2893 a 1072 2947 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3.2) cvn H.B /ANN pdfmark end 1072 2947 a FD(\))h(in)n(to)f(\()1333 2947 y SDict begin H.S end 1333 2947 a Black FD(3)p Black 1374 2893 a SDict begin H.R end 1374 2893 a 1374 2947 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3.3) cvn H.B /ANN pdfmark end 1374 2947 a FD(\))h(w)n(e)g(get)f(the)h(lemma.)1246 b Fs(u)-55 b(t)648 3093 y FD(Appropriately)31 b(c)n(ho)r(osing)g(times)i(\(rather) g(than)h(n)n(um)n(b)r(er)f(of)h(elemen)n(ts\))f(as)g(a)h(func-)523 3192 y(tion)29 b(of)g(the)h(prior)d(distribution)g(allo)n(ws)f(us)j(to) g(do)h(the)f(same)f(for)h(other)g(v)-5 b(arian)n(ts)27 b(of)j(the)523 3292 y(secretary)h(problem,)g(the)j(Knapsac)n(k)d(\(ac)n (hieving)f(a)j(comp)r(etitiv)n(e)e(ratio)g(of)3014 3259 y FC(1)p 3014 3273 34 4 v 3014 3321 a(2)3079 3292 y Fs(\001)3167 3259 y FC(1)p 3134 3273 98 4 v 3134 3321 a(10)p Ft(e)3242 3292 y FD(,)i(see)523 3392 y([)546 3392 y SDict begin H.S end 546 3392 a Black FD(2)p Black 588 3338 a SDict begin H.R end 588 3338 a 588 3392 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff07-knapsack) cvn H.B /ANN pdfmark end 588 3392 a FD(]\))28 b(and)f(Matrio)r(d)f(\()p Fv(O)r FD(\(ln)14 b(ln)f Fv(\032)p FD(\))28 b(when)g Fv(\032)g FD(is)e(the)i(rank)f(of)g (the)h(Matroid,)e(see)h([)2998 3392 y SDict begin H.S end 2998 3392 a Black FD(26)p Black 3081 3338 a SDict begin H.R end 3081 3338 a 3081 3392 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lachish14) cvn H.B /ANN pdfmark end 3081 3392 a FD(,)3104 3392 y SDict begin H.S end 3104 3392 a Black FD(14)p Black 3188 3338 a SDict begin H.R end 3188 3338 a 3188 3392 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.FeldmanSZ15) cvn H.B /ANN pdfmark end 3188 3392 a FD(]\).)523 3540 y SDict begin H.S end 523 3540 a 523 3540 a SDict begin 12 H.A end 523 3540 a 523 3540 a SDict begin [/View [/XYZ H.V]/Dest (section.1.4) cvn /DEST pdfmark end 523 3540 a 95 x Fx(4)112 b(Impro)m(v)m(ed)41 b(results)f(for)g(the)g(temp)g (secretary)g(problem)h(for)691 3751 y(the)d(uniform)g(arriv)-6 b(al)38 b(distribution)523 3928 y FD(In)26 b(this)f(section)g(w)n(e)g (giv)n(e)f(an)i(impro)n(v)n(ed)d(algorithm,)e(referred)k(as)g(the)i(c)n (harter)d(algorithm)523 4028 y Fv(C)582 4040 y Ft(k)q(;\015)681 4028 y FD(,)k(for)f(the)h(temp)f(secretary)f(problem)f(with)i(uniform)f (arriv)-5 b(al)24 b(times)i(and)h(capacit)n(y)e(1)523 4127 y(\(at)j(most)e(one)h(secretary)f(can)h(b)r(e)h(hired)f(at)g(an)n (y)g(time\).)648 4227 y(As)e(it)f(is)g(nev)n(er)g(the)i(case)e(that)h (more)f(than)h(1)p Fv(=\015)k FD(items)23 b(can)i(b)r(e)g(selected,)f (setting)g Fv(k)i FD(=)523 4327 y Fs(d)p FD(1)p Fv(=\015)t Fs(e)33 b FD(e\013ectiv)n(ely)f(remo)n(v)n(es)f(the)j(budget)g (constrain)n(t.)d(Note)j(that)g Fv(C)2781 4339 y Ft(k)q(;)p FC(0)2909 4327 y FD(is)e(Klein)n(b)r(erg's)523 4426 y(algorithm)27 b(for)k(the)h Fv(k)s FD(-secretary)d(problem,)g(with)i(some)f(missing)e (details)h(added)i(to)h(the)523 4526 y(description.)648 4625 y(T)-7 b(o)23 b(analyze)e(the)j(c)n(harter)d(algorithm)f(w)n(e)j (establish)d(a)j(lo)n(w)n(er)e(b)r(ound)j(on)f(the)h(exp)r(ected)523 4725 y(size)j(of)h(the)g(maxim)n(um)c Fv(\015)5 b FD(-indep)r(enden)n (t)28 b(subset)g(of)g(a)g(set)g(of)g(uniformly)d(random)h(p)r(oin)n(ts) 523 4825 y(in)h([0)p Fv(;)14 b FD(1\).)28 b(W)-7 b(e)29 b(apply)d(this)i(lo)n(w)n(er)e(b)r(ound)i(to)g(the)h(subset)f(of)g(the) g(items)f(that)h(Klein)n(b)r(erg's)523 4924 y(algorithm)23 b(selects.)p Black 1943 5173 a(6)p Black eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 515 125 a Black Black 523 365 a SDict begin H.S end 523 365 a 523 365 a SDict begin 12 H.A end 523 365 a 523 365 a SDict begin [/View [/XYZ H.V]/Dest (subsection.1.4.1) cvn /DEST pdfmark end 523 365 a 83 x Fr(4.1)95 b(The)35 b(temp)f(secretary)j(algorithm,)c Fm(C)2138 460 y Fl(k;\015)2247 448 y Fr(:)i(a)h(comp)s(etitiv)m(e)e (ratio)h(of)741 548 y(1)p Fm(=)p Fr(\(1)20 b(+)i Fm(k)r(\015)5 b Fr(\).)523 712 y FD(This)26 b(c)n(harter)f(algorithm,)e Fv(C)1455 724 y Ft(k)q(;\015)1581 712 y FD(gets)k(parameters)d Fv(k)30 b FD(\(the)d(maximal)c(n)n(um)n(b)r(er)i(of)i(ren)n(tals)523 811 y(allo)n(w)n(ed\))h(and)j Fv(\015)36 b FD(\(the)c(ren)n(tal)d(p)r (erio)r(d\))i(as)f(is)g(describ)r(ed)g(in)h(detail)e(in)h(Algorithm) 3214 811 y SDict begin H.S end 3214 811 a Black FD(1)p Black 3256 758 a SDict begin H.R end 3256 758 a 3256 811 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (algocfline.1) cvn H.B /ANN pdfmark end 3256 811 a FD(.)h(As)523 911 y(the)i(en)n(tire)e(p)r (erio)r(d)h(is)g(normalized)d(to)j([0)p Fv(;)14 b FD(1\),)32 b(ha)n(ving)f Fv(k)j(>)e Fs(d)p FD(1)p Fv(=\015)s Fs(e)h FD(is)f(irrelev)-5 b(an)n(t.)29 b(Th)n(us,)523 1010 y(w)n(e)e(assume)f (that)i Fv(k)e Fs(\024)d(d)o FD(1)p Fv(=\015)t Fs(e)p FD(.)1496 1010 y SDict begin H.S end 1496 1010 a -30 x FC(2)1533 1010 y SDict begin 12 H.L end 1533 1010 a 1533 1010 a SDict begin [/Subtype /Link/Dest (Hfootnote.2) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 1533 1010 a 648 1110 a FD(W)-7 b(e)32 b(sho)n(w)g(that)h Fv(C)1250 1122 y Ft(k)q(;\015)1349 1110 y FD(\()p Fv(X)7 b FD(\))33 b(gains)d(in)i(exp)r(ectation)f(ab)r (out)i(1)p Fv(=)p FD(\(1)21 b(+)g Fv(k)s(\015)5 b FD(\))32 b(of)h(the)g(top)f Fv(k)523 1210 y FD(v)-5 b(alues)24 b(of)h Fv(X)7 b FD(,)25 b(whic)n(h)f(implies)d(that)26 b(the)f(comp)r(etitiv)n(e)e(ratio)g(\(see)i(de\014nition)f(\()3023 1210 y SDict begin H.S end 3023 1210 a Black FD(1)p Black 3065 1156 a SDict begin H.R end 3065 1156 a 3065 1210 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2.1) cvn H.B /ANN pdfmark end 3065 1210 a FD(\)\))h(of)h Fv(C)3306 1222 y Ft(k)q(;\015)523 1309 y FD(is)h(at)g(least)f(ab)r(out)i(1)p Fv(=)p FD(\(1)17 b(+)h Fv(k)s(\015)5 b FD(\).)648 1409 y(Note)27 b(that)g(for)g Fv(k)f FD(=)d Fs(d)o FD(1)p Fv(=\015)t Fs(e)p FD(,)k Fv(C)1624 1421 y Ft(k)q(;\015)1751 1409 y FD(has)g(a)f(comp)r(etitiv)n(e)f(ratio)g(close)g(to)i(1)p Fv(=)p FD(2,)f(while)g(for)523 1509 y Fv(\015)i FD(=)22 b(0,)28 b Fv(C)833 1521 y Ft(k)q(;\015)960 1509 y FD(has)f(a)g(comp)r (etitiv)n(e)e(ratio)g(close)h(to)h(1.)523 1532 y SDict begin H.S end 523 1532 a 523 1532 a SDict begin 12 H.A end 523 1532 a 523 1532 a SDict begin [/View [/XYZ H.V]/Dest (algocfline.1) cvn /DEST pdfmark end 523 1532 a 648 1608 a FD(It)h(is)e(easy)h(to)g(see)g(that)h Fv(C)1478 1620 y Ft(k)q(;\015)1578 1608 y FD(,)f(c)n(ho)r(oses)f(a)i Fv(\015)5 b FD(-indep)r(enden)n(t)26 b(set)i(of)f(size)g(at)g(most)g Fv(k)s FD(.)648 1708 y(The)c(main)f(theorem)g(of)i(this)f(pap)r(er)g (is)g(the)h(follo)n(wing)c(generalization)f(of)k(Klein)n(b)r(erg's)523 1807 y Fv(k)s FD(-secretary)i(problem:)523 1824 y SDict begin H.S end 523 1824 a 523 1824 a SDict begin 12 H.A end 523 1824 a 523 1824 a SDict begin [/View [/XYZ H.V]/Dest (theorem.1.4.2) cvn /DEST pdfmark end 523 1824 a Black 155 x Fr(Theorem)30 b(2.)p Black 41 w Fp(F)-6 b(or)36 b(any)g(set)f(of)h(items)g Fv(S)i FD(=)33 b Fs(f)p Fv(x)2113 1991 y Ft(i)2141 1979 y Fs(g)2183 1949 y Ft(n)2183 2000 y(i)p FC(=1)2294 1979 y Fp(,)j FD(0)d Fv(<)g(\015)38 b Fs(\024)33 b Fv(\015)2755 1949 y Fq(\003)2826 1979 y FD(=)g(0)p Fv(:)p FD(003176)f Fp(and)523 2078 y(any)e(p)l(ositive)h (inte)l(ger)f Fv(k)c Fs(\024)d FD(1)p Fv(=\015)5 b Fp(:)1011 2297 y SDict begin H.S end 1011 2297 a 1011 2297 a SDict begin 12 H.A end 1011 2297 a 1011 2297 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.4.4) cvn /DEST pdfmark end 1011 2297 a FD(E)1068 2312 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1])1357 2297 y FD([)p Fv(v)s FD(\()p Fv(C)1514 2309 y Ft(k)q(;\015)1614 2297 y FD(\()p Fv(S;)14 b(\022)r FD(\)\)])25 b Fs(\025)2082 2241 y FD(1)p 1984 2278 237 4 v 1984 2354 a(1)18 b(+)g Fv(\015)5 b(k)2231 2297 y FD(\(1)18 b Fs(\000)g Fv(\014)t FD(\()p Fv(\015)5 b(;)14 b(k)s FD(\)\))p Fv(T)2733 2309 y Ft(k)2774 2297 y FD(\()p Fv(S)5 b FD(\))p Fv(;)382 b FD(\(4\))523 2536 y Fp(wher)l(e)31 b Fv(\014)t FD(\()p Fv(\015)5 b(;)14 b(k)s FD(\))25 b(=)f(7)p Fv(:)p FD(4)1225 2465 y Fu(p)p 1307 2465 326 4 v 1307 2536 a Fv(\015)18 b FD(ln)o(\(1)p Fv(=\015)5 b FD(\))19 b(+)1774 2503 y FC(5)p 1745 2517 92 4 v 1745 2526 a Fq(p)p 1800 2526 37 3 v 50 x Ft(k)1846 2536 y Fv(;)31 b Fp(and)g(the)f(exp)l(e)l(ctation)h(is)g(taken)f(oven)h (al)t(l)h(uni-)523 2652 y(form)43 b(mappings)h(of)f Fv(S)k Fp(to)42 b(the)h(interval)g FD([0)p Fv(;)14 b FD(1\))p Fp(.)42 b(\(Note)g(that)g(the)h(right)g(hand)g(side)g(of)523 2751 y(Equation)30 b(\()910 2768 y SDict begin H.S end 910 2768 a Black -17 x Fp(4)p Black 953 2698 a SDict begin H.R end 953 2698 a 953 2751 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.4.4) cvn H.B /ANN pdfmark end 953 2751 a Fp(\))f(is)i(ne)l(gative)f(for)h Fv(\015)1607 2721 y Fq(\003)1667 2751 y Fv(<)23 b(\015)28 b Fs(\024)22 b FD(0)p Fv(:)p FD(5)p Fp(.\))523 2919 y SDict begin H.S end 523 2919 a 523 2919 a SDict begin 12 H.A end 523 2919 a 523 2919 a SDict begin [/View [/XYZ H.V]/Dest (subsection.1.4.2) cvn /DEST pdfmark end 523 2919 a 79 x Fr(4.2)95 b(Outline)30 b(of)i(the)g(pro)s(of)f(of)h(Theorem)2124 2998 y SDict begin H.S end 2124 2998 a Black Fr(2)p Black 2172 2945 a SDict begin H.R end 2172 2945 a 2172 2998 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.2) cvn H.B /ANN pdfmark end 2172 2998 a 523 3162 a FD(W)-7 b(e)23 b(pro)n(v)n(e)e(Theorem)1225 3162 y SDict begin H.S end 1225 3162 a Black FD(2)p Black 1267 3108 a SDict begin H.R end 1267 3108 a 1267 3162 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.2) cvn H.B /ANN pdfmark end 1267 3162 a 22 w FD(b)n(y)h(induction)f(on)h Fv(k)s FD(.)g(F)-7 b(or)22 b Fv(k)k Fs(\024)d FD(25)e(the)i(theorem)e(holds)g(v)-5 b(acuously)e(.)648 3261 y(The)21 b(pro\014t,)f Fv(p)1096 3231 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1298 3261 y FD(,)h(on)f(those)h(items)e(that)i(arriv)n(e)d(during)i(the)h(time)e (in)n(terv)-5 b(al)19 b([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))523 3361 y(is)31 b(giv)n(en)g(b)n(y)h(the)h(inductiv)n(e)e(h)n (yp)r(othesis)1849 3361 y SDict begin H.S end 1849 3361 a -30 x FC(3)1884 3361 y SDict begin 12 H.L end 1884 3361 a 1884 3361 a SDict begin [/Subtype /Link/Dest (Hfootnote.3) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 1884 3361 a FD(.)i(Ho)n(w)n(ev)n(er,)e (the)i(inductiv)n(e)d(h)n(yp)r(othesis)h(giv)n(es)523 3461 y(this)h(pro\014t,)h Fv(p)987 3431 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1188 3461 y FD(,)g(in)f(terms)g(of)g(the)h(top)g Fs(b)p Fv(k)s(=)p FD(2)p Fs(c)f FD(elemen)n(ts)e(that)j(arriv)n(e)d(b)r (efore)i(time)523 3560 y(1)p Fv(=)p FD(2,)d(and)h(not)g(in)f(terms)g (of)h Fv(T)1494 3572 y Ft(k)1535 3560 y FD(\()p Fv(X)7 b FD(\),)30 b(the)g(v)-5 b(alue)29 b(of)h(the)h(top)f Fv(k)j FD(items)28 b(o)n(v)n(erall.)e(Th)n(us,)k(w)n(e)523 3660 y(need)e(to)g(relate)f Fv(p)1093 3630 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1323 3660 y FD(to)h Fv(T)1474 3672 y Ft(k)1514 3660 y FD(\()p Fv(X)7 b FD(\).)28 b(In)h(Lemma)2107 3660 y SDict begin H.S end 2107 3660 a Black FD(2)p Black 2148 3606 a SDict begin H.R end 2148 3606 a 2148 3660 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.2) cvn H.B /ANN pdfmark end 2148 3660 a 29 w FD(w)n(e)e(sho)n(w)h(that)g Fv(p)2728 3630 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2958 3660 y FD(is)f(ab)r(out)h(1/2)523 3760 y(of)g Fv(T)667 3772 y Ft(k)707 3760 y FD(\()p Fv(X)7 b FD(\).)648 3859 y(Let)25 b Fv(Z)851 3871 y Ft(>T)980 3859 y FD(b)r(e)g(the)h(set)f(of)g (items)e(that)j(arriv)n(e)c(in)i(the)i(time)d(in)n(terv)-5 b(al)23 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))23 b(and)i(ha)n(v)n(e)523 3959 y(v)-5 b(alue)24 b(greater)f(than)j(the)f(threshold)f Fv(T)12 b FD(.)24 b(F)-7 b(rom)24 b Fv(Z)2090 3971 y Ft(>T)2219 3959 y FD(w)n(e)g(greedily)e(pic)n(k)i(a)h Fv(\015)5 b FD(-indep)r(enden)n(t)523 4058 y(subset)750 4058 y SDict begin H.S end 750 4058 a -30 x FC(4)787 4058 y SDict begin 12 H.L end 787 4058 a 787 4058 a SDict begin [/Subtype /Link/Dest (Hfootnote.4) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 787 4058 a FD(.)25 b(It)g(is)f(easy)f(to)i(see)f(that)h(this)f (set)h(is)e(in)h(fact)h(a)f(maximal)c Fv(\015)5 b FD(-indep)r(enden)n (t)24 b(subset.)648 4158 y(T)-7 b(o)26 b(b)r(ound)i(the)g(exp)r(ected)f (pro\014t)g(from)f(the)i(items)d(in)i Fv(Z)2453 4170 y Ft(>T)2584 4158 y FD(w)n(e)g(\014rst)g(b)r(ound)h(the)f(size)523 4258 y(of)37 b(the)g(maximal)32 b Fv(\015)5 b FD(-indep)r(enden)n(t)36 b(set)h(amongst)e(these)h(items.)g(T)-7 b(o)36 b(do)g(so)h(w)n(e)f(use) h(the)523 4357 y(follo)n(wing)24 b(general)h(theorem)h(\(see)h(also)f (Section)2106 4357 y SDict begin H.S end 2106 4357 a Black FD(6)p Black 2148 4304 a SDict begin H.R end 2148 4304 a 2148 4357 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.1.6) cvn H.B /ANN pdfmark end 2148 4357 a 28 w FD(and)h(Section)2628 4357 y SDict begin H.S end 2628 4357 a Black FD(C)p Black 2688 4301 a SDict begin H.R end 2688 4301 a 2688 4357 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.A.3) cvn H.B /ANN pdfmark end 2688 4357 a FD(\).)523 4378 y SDict begin H.S end 523 4378 a 523 4378 a SDict begin 12 H.A end 523 4378 a 523 4378 a SDict begin [/View [/XYZ H.V]/Dest (theorem.1.4.3) cvn /DEST pdfmark end 523 4378 a Black 523 4430 473 4 v 546 4484 a Fn(2)606 4515 y SDict begin H.S end 606 4515 a 606 4515 a SDict begin H.R end 606 4515 a 606 4515 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.2) cvn /DEST pdfmark end 606 4515 a FB(T)-6 b(o)26 b(simplify)g(the)f(presen)n(tation)h(w)n(e)g(shall)h(assume)e (the)g(in)h(sequel)g(that)f Fy(k)f Fk(\024)d FB(1)p Fy(=\015)t FB(.)546 4597 y Fn(3)606 4629 y SDict begin H.S end 606 4629 a 606 4629 a SDict begin H.R end 606 4629 a 606 4629 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.3) cvn /DEST pdfmark end 606 4629 a FB(This)38 b(pro\014t,)f Fy(p)1074 4597 y Fn([0)p Fj(;)p Fn(1)p Fj(=)p Fn(2\))1298 4629 y FB(is)g(E)1438 4641 y Fj(\022)r Fn(:)p Fj(S)s Fi(7!)p Fn([0)p Fj(;)p Fn(1\))1714 4546 y Fh(h)1750 4629 y Fy(v)s FB(\()p Fy(C)1880 4591 y Fn([0)p Fj(;)p Fn(1)p Fj(=)p Fn(2\))1875 4650 y Fj(k)q(;\015)2068 4629 y FB(\()p Fy(S;)13 b(\022)r FB(\)\))2277 4546 y Fh(i)2313 4629 y FB(,)37 b(where)h Fy(C)2665 4591 y Fn([0)p Fj(;)p Fn(1)p Fj(=)p Fn(2\))2660 4650 y Fj(k)q(;\015)2852 4629 y FB(\()p Fy(S;)13 b(\022)r FB(\))37 b(the)g(set)g(of)606 4742 y(items)25 b(c)n(hosen)h(b)n(y)f(the)g(algorithm)h(during)f(the)h(time) f(p)r(erio)r(d)h([0)p Fy(;)14 b FB(1)p Fy(=)p FB(2\).)546 4801 y Fn(4)606 4833 y SDict begin H.S end 606 4833 a 606 4833 a SDict begin H.R end 606 4833 a 606 4833 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.4) cvn /DEST pdfmark end 606 4833 a FB(mo)r(dulo)31 b(the)g(ca)n(v)n(eat)h(that)f (the)g(arriv)l(al)h(time)e(of)j(the)e(1st)h(item)e(c)n(hosen)i(from)f (the)g(2nd)g(half)606 4924 y(m)n(ust)24 b(b)r(e)i(at)g(least)g Fy(\015)k FB(after)c(the)g(arriv)l(al)g(time)f(of)i(the)e(last)h(item)f (c)n(hosen)h(in)g(the)f(1st)h(half.)p Black Black 1943 5173 a FD(7)p Black eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 515 125 a Black Black Black 523 397 2882 7 v Black 523 465 a Fz(ALGORITHM)30 b(1:)e FB(The)e(Charter)g(Algorithm)f Fy(C)2087 474 y Fj(k)q(;\015)2180 465 y FB(.)p Black 523 507 V Black 523 524 a SDict begin H.S end 523 524 a 523 524 a SDict begin 11 H.A end 523 524 a 523 524 a SDict begin [/View [/XYZ H.V]/Dest (algocf.1) cvn /DEST pdfmark end 523 524 a 523 588 a SDict begin H.S end 523 588 a 523 588 a SDict begin 11 H.A end 523 588 a 523 588 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.1) cvn /DEST pdfmark end 523 588 a -79 w Fg(1)41 b Fz(if)34 b Fy(k)23 b FB(=)e(1)26 b Fz(then)p 565 980 4 366 v Black 651 679 a FA(/*)p Black 40 w(Use)39 b(the)h(``continuous)i(secretary'')g(algorithm)f([)2494 679 y SDict begin H.S end 2494 679 a Black FA(13)p Black 2573 633 a SDict begin H.R end 2573 633 a 2573 679 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.FNS11) cvn H.B /ANN pdfmark end 2573 679 a FA(]:)672 b(*/)651 770 y SDict begin H.S end 651 770 a 651 770 a SDict begin 11 H.A end 651 770 a 651 770 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.2) cvn /DEST pdfmark end 651 770 a -207 w Fg(2)169 b FB(Let)26 b Fy(x)f FB(b)r(e)g(the)g(largest)j(item)d(to)g (arriv)n(e)h(b)n(y)f(time)g(1)p Fy(=e)h FB(\(if)g(no)g(item)f(arriv)n (es)h(b)n(y)f(time)g(1)p Fy(=e)h FB(|)651 862 y(let)g Fy(x)f FB(b)r(e)h(the)f(absolute)h(zero,)h(an)e(item)g(smaller)i(than)e (all)h(other)g(items\).)651 953 y SDict begin H.S end 651 953 a 651 953 a SDict begin 11 H.A end 651 953 a 651 953 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.3) cvn /DEST pdfmark end 651 953 a -207 w Fg(3)169 b Fy(C)706 962 y Fj(k)q(;\015)824 953 y FB(accepts)26 b(the)g(\014rst)f(item)g Fy(y)s FB(,)g Fy(y)f(>)d(x)p FB(,)26 b(that)f(arriv)n(es)h(after)h (time)e(1)p Fy(=e)h FB(\(if)g(an)n(y\))523 1053 y SDict begin H.S end 523 1053 a 523 1053 a SDict begin 11 H.A end 523 1053 a 523 1053 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.4) cvn /DEST pdfmark end 523 1053 a -79 w Fg(4)41 b Fz(else)p 565 3352 4 2272 v Black 651 1144 a FA(/*)p Black 40 w(Process)f(the)g(items)h(scheduled)g(during)g(the)e (time)i(interval)g FB([0)p Fy(;)26 b FB(1)p Fy(=)p FB(2\))225 b FA(*/)651 1235 y SDict begin H.S end 651 1235 a 651 1235 a SDict begin 11 H.A end 651 1235 a 651 1235 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.5) cvn /DEST pdfmark end 651 1235 a -207 w Fg(5)169 b FB(Initiate)26 b(a)g(recursiv)n(e)g (cop)n(y)f(of)h(the)g(algorithm,)g Fy(C)2137 1203 y Fi(0)2181 1235 y FB(=)21 b Fy(C)2317 1247 y Fi(b)o Fj(k)q(=)p Fn(2)p Fi(c)p Fj(;)p Fn(2)p Fj(\015)2559 1235 y FB(.)651 1327 y SDict begin H.S end 651 1327 a 651 1327 a SDict begin 11 H.A end 651 1327 a 651 1327 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.6) cvn /DEST pdfmark end 651 1327 a -207 w Fg(6)169 b Fy(x)21 b Fk( )k FB(next)g(elemen)n(t)p Black 773 w FA(//)p Black 40 w(If)40 b(no)f(further)i(items)g(arrive)f Fy(x)21 b Fk( )g(;)651 1418 y SDict begin H.S end 651 1418 a 651 1418 a SDict begin 11 H.A end 651 1418 a 651 1418 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.7) cvn /DEST pdfmark end 651 1418 a -207 w Fg(7)169 b Fz(while)24 b Fy(x)d Fk(6)p FB(=)g Fk(;)55 b Ff(AND)g Fy(\022)r FB(\()p Fy(x)p FB(\))21 b Fy(<)g FB(1)p Fy(=)p FB(2)27 b Fz(do)p 692 1846 4 401 v 779 1509 a SDict begin H.S end 779 1509 a 779 1509 a SDict begin 11 H.A end 779 1509 a 779 1509 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.1.8) cvn /DEST pdfmark end 779 1509 a -335 w Fg(8)297 b FB(Sim)n(ulate)25 b Fy(C)1157 1477 y Fi(0)1205 1509 y FB(with)h(input)e Fy(x)i FB(and)f(mo)r(di\014ed)f(sc)n(hedule)i Fy(\022)2460 1477 y Fi(0)2482 1509 y FB(\()p Fy(x)p FB(\))21 b(=)g(2)p Fy(\022)r FB(\()p Fy(x)p FB(\).)779 1601 y SDict begin H.S end 779 1601 a 779 1601 a 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1637 4013 423 4 v 1728 4089 a FD(1)18 b(+)g Fv(n\015)2070 4032 y(;)98 b FD(where)27 b Fv(\013)p FD(\()p Fv(\015)5 b FD(\))24 b(=)f(3)2750 3957 y Fu(p)p 2832 3957 326 4 v 2832 4032 a Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))o Fv(;)118 b FD(\(5\))523 4264 y Fp(wher)l(e)30 b Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))30 b Fp(denotes)g(the)g(size)g(of)h(the)f (lar)l(gest)g Fv(\015)5 b Fp(-indep)l(endent)30 b(subset)f(of)h Fv(Z)6 b Fp(.)648 4441 y FD(W)-7 b(e)28 b(apply)f(Theorem)1371 4441 y SDict begin H.S end 1371 4441 a Black FD(3)p Black 1413 4387 a SDict begin H.R end 1413 4387 a 1413 4441 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 1413 4441 a 28 w FD(to)h(the)g(items)f(in)g Fv(Z)2063 4453 y Ft(>T)2167 4441 y FD(.)h(W)-7 b(e)29 b(can)e(apply)g(this)h(theorem)e(since)523 4540 y(arriv)-5 b(al)26 b(times)h(of)j(items)d(in)i Fv(Z)1485 4552 y Ft(>T)1618 4540 y 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(al)j(Computer)f(Scienc)l(e)p FB(,)f(130\(1\):5)h({)e(16,)h(1994.)p Black 1922 5173 a FD(12)p Black eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a SDict begin H.S end 523 448 a 523 448 a SDict begin 14 H.A end 523 448 a 523 448 a SDict begin [/View [/XYZ H.V]/Dest (section*.3) cvn /DEST pdfmark end 523 448 a Fx(APPENDIX)523 651 y SDict begin H.S end 523 651 a 523 651 a SDict begin 12 H.A end 523 651 a 523 651 a SDict begin [/View [/XYZ H.V]/Dest (section.A.1) cvn /DEST pdfmark end 523 651 a 142 x Fx(A)112 b(T)-9 b(able)38 b(of)f(results)p Black Black 718 1288 a Fz(T)-7 b(able)27 b(1.)1053 1288 y SDict begin H.S end 1053 1288 a 1053 1288 a SDict begin H.R end 1053 1288 a 1053 1288 a SDict begin [/View [/XYZ H.V]/Dest (table.A.1) cvn /DEST pdfmark end 1053 1288 a FB(Size)f(of)g(maximal)f Fy(\015)t FB(-indep)r(enden)n(t)e(set.)k(Uniformly)e(prior)h(on)f (arriv)l(als.)p Black 523 1397 2680 4 v 521 1488 4 92 v 1167 1488 V 2225 1488 V 3201 1488 V 521 1762 4 274 v 546 1552 a(Size)h(of)h(maximal)546 1643 y Fy(\015)t FB(-indep)r(enden)n(t)d(set)546 1735 y(capacit)n(y)i Fy(d)21 b FB(=)g(1)p 1167 1762 V 1256 1613 a Fj(n)p 1190 1627 170 4 v 1190 1669 a Fn(1+)p Fj(\015)s Fi(\001)p Fj(n)1370 1561 y Fh(\020)1416 1644 y FB(1)c Fk(\000)g Fy(\002)1608 1561 y Fh(\020)1655 1580 y(p)p 1732 1580 302 4 v 64 x Fy(\015)f FB(ln\(1)p Fy(=\015)t FB(\))2033 1561 y Fh(\021\021)p 2225 1762 4 274 v 2238 1644 a FB(Theorem)2562 1644 y SDict begin H.S end 2562 1644 a Black FB(3)p Black 2601 1596 a SDict begin H.R end 2601 1596 a 2601 1644 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2601 1644 a 3201 1762 4 274 v 521 1853 4 92 v 1167 1853 V 2225 1853 V 3201 1853 V 523 1857 2680 4 v 521 1948 4 92 v 1167 1948 V 2225 1948 V 3201 1948 V 521 2337 4 389 v 546 2024 a FB(Size)26 b(of)h(maximal)546 2115 y Fy(\015)t FB(-indep)r(enden)n(t)d(set)546 2207 y(of)j(capacit)n(y)f Fy(d)546 2298 y(n)c FB(=)f Fy(d)c Fk(\001)g FB(1)p Fy(=\015)t FB(.)p 1167 2337 V 1192 2034 a Fk(\025)k Fy(n)1319 1951 y Fh(\020)1365 2034 y FB(1)d Fk(\000)e Fy(\002)1557 1951 y Fh(\020)1614 1961 y Fi(p)p 1664 1961 94 3 v 42 x Fn(ln)10 b Fj(d)p 1614 2017 144 4 v 1645 2025 a Fi(p)p 1694 2025 33 3 v 1694 2068 a Fj(d)1767 1951 y Fh(\021\021)1192 2289 y Fk(\024)21 b Fy(n)1319 2206 y Fh(\020)1365 2289 y FB(1)d Fk(\000)e Fy(\002)1557 2206 y Fh(\020)1639 2258 y Fn(1)p 1614 2272 82 4 v 1614 2280 a Fi(p)p 1664 2280 33 3 v 42 x Fj(d)1706 2206 y Fh(\021)o(\021)p 2225 2337 4 389 v 2250 2115 a FB(Lo)n(w)n(er)26 b(b)r(ound)f(|)g(Theorem)3141 2115 y SDict begin H.S end 3141 2115 a Black FB(5)p Black 3179 2067 a SDict begin H.R end 3179 2067 a 3179 2115 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.A.4.5) cvn H.B /ANN pdfmark end 3179 2115 a 2250 2207 a FB(Lo)n(w)n(er)h(b)r(ound)f(|)g(Theorem)3141 2207 y SDict begin H.S end 3141 2207 a Black FB(6)p Black 3179 2158 a SDict begin H.R end 3179 2158 a 3179 2207 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.A.4.6) cvn H.B /ANN pdfmark end 3179 2207 a 3201 2337 4 389 v 521 2428 4 92 v 1167 2428 V 2225 2428 V 3201 2428 V 523 2432 2680 4 v 521 2523 4 92 v 1167 2523 V 2225 2523 V 3201 2523 V 521 2797 4 274 v 546 2587 a FB(Size)h(of)h(maximal)546 2678 y Fy(\015)t FB(-indep)r(enden)n(t)d(set)546 2770 y(of)j(capacit)n(y)f Fy(d)p 1167 2797 V 1192 2633 a Fk(\025)21 b FB(min)o(\()p Fy(n;)13 b(d=\015)t FB(\))1663 2550 y Fh(\020)1709 2633 y FB(1)18 b Fk(\000)e Fy(\002)1901 2550 y Fh(\020)1958 2560 y Fi(p)p 2008 2560 94 3 v 42 x Fn(ln)10 b Fj(d)p 1958 2616 144 4 v 1989 2624 a Fi(p)p 2038 2624 33 3 v 2038 2667 a Fj(d)2111 2550 y Fh(\021\021)1192 2746 y Fk(\024)21 b FB(min)o(\()p Fy(n;)13 b(d=\015)t FB(\))p 2225 2797 4 274 v 2238 2679 a(Theorem)2562 2679 y SDict begin H.S end 2562 2679 a Black FB(5)p Black 2601 2631 a SDict begin H.R end 2601 2631 a 2601 2679 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.A.4.5) cvn H.B /ANN pdfmark end 2601 2679 a 3201 2797 4 274 v 521 2888 4 92 v 1167 2888 V 2225 2888 V 3201 2888 V 523 2892 2680 4 v Black Black Black 1026 3399 a Fz(T)-7 b(able)28 b(2.)1361 3399 y SDict begin H.S end 1361 3399 a 1361 3399 a SDict begin H.R end 1361 3399 a 1361 3399 a SDict begin [/View [/XYZ H.V]/Dest (table.A.2) cvn /DEST pdfmark end 1361 3399 a FB(Comp)r(etitiv)n(e)e(ratios,)h(arbitrary)f(prior)g(on)g (arriv)l(als)p Black 523 3508 2272 4 v 521 3599 4 92 v 1748 3599 V 2176 3599 V 2793 3599 V 521 3691 V 546 3663 a(Capacit)n(y)h(one)e Fy(\015)t FB(-indep)r(enden)n(t)f(subset)p 1748 3691 V 1786 3633 a Fn(1)p 1771 3647 60 4 v 1771 3689 a(2)p Fj(e)p 2176 3691 4 92 v 2190 3664 a FB(Theorem)2514 3664 y SDict begin H.S end 2514 3664 a Black FB(1)p Black 2552 3615 a SDict begin H.R end 2552 3615 a 2552 3664 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.3.1) cvn H.B /ANN pdfmark end 2552 3664 a 2793 3691 4 92 v 521 3782 V 1748 3782 V 2176 3782 V 2793 3782 V 523 3786 2272 4 v 521 3877 4 92 v 1748 3877 V 2176 3877 V 2793 3877 V 521 3969 V 546 3941 a FB(Matroid)j(constrain)n(ts)p 1748 3969 V 553 w Fy(\012)t FB(\(1)p Fy(=)13 b FB(ln)g(ln)g Fy(\032)p FB(\))p 2176 3969 V 23 w([)2211 3941 y SDict begin H.S end 2211 3941 a Black FB(26)p Black 2288 3893 a SDict begin H.R end 2288 3893 a 2288 3941 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Lachish14) cvn H.B /ANN pdfmark end 2288 3941 a FB(])26 b(&)f(Theorem)2744 3941 y SDict begin H.S end 2744 3941 a Black FB(1)p Black 2783 3893 a SDict begin H.R end 2783 3893 a 2783 3941 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.3.1) cvn H.B /ANN pdfmark end 2783 3941 a 2793 3969 4 92 v 521 4060 V 1748 4060 V 2176 4060 V 2793 4060 V 523 4063 2272 4 v 521 4155 4 92 v 1748 4155 V 2176 4155 V 2793 4155 V 521 4246 V 546 4219 a FB(Knapsac)n(k)h(constrain)n(ts)p 1748 4246 V 1801 4188 a Fn(1)p 1771 4202 91 4 v 1771 4244 a(20)p Fj(e)p 2176 4246 4 92 v 2190 4219 a FB([)2211 4219 y SDict begin H.S end 2211 4219 a Black FB(2)p Black 2249 4171 a SDict begin H.R end 2249 4171 a 2249 4219 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Babaioff07-knapsack) cvn H.B /ANN pdfmark end 2249 4219 a FB(])g(&)g(Theorem)2706 4219 y SDict begin H.S end 2706 4219 a Black FB(1)p Black 2744 4171 a SDict begin H.R end 2744 4171 a 2744 4219 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.3.1) cvn H.B /ANN pdfmark end 2744 4219 a 2793 4246 4 92 v 521 4338 V 1748 4338 V 2176 4338 V 2793 4338 V 523 4341 2272 4 v 521 4432 4 92 v 1748 4432 V 2176 4432 V 2793 4432 V 521 4615 4 183 v 546 4496 a FB(Capacit)n(y)h Fy(d)546 4588 y(\015)t FB(-indep)r(enden)n(t)d(set)p 1748 4615 V 1771 4512 a Fn(1)p 1771 4526 31 4 v 1771 4568 a(2)1811 4460 y Fh(\020)1857 4542 y FB(1)18 b Fk(\000)2025 4512 y Fn(5)p 1999 4526 82 4 v 1999 4534 a Fi(p)p 2049 4534 33 3 v 42 x Fj(d)2091 4460 y Fh(\021)p 2176 4615 4 183 v 2190 4542 a FB([)2211 4542 y SDict begin H.S end 2211 4542 a Black FB(24)p Black 2288 4494 a SDict begin H.R end 2288 4494 a 2288 4542 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kleinberg) cvn H.B /ANN pdfmark end 2288 4542 a FB(])26 b(&)f(Theorem)2744 4542 y SDict begin H.S end 2744 4542 a Black FB(1)p Black 2783 4494 a SDict begin H.R end 2783 4494 a 2783 4542 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.3.1) cvn H.B /ANN pdfmark end 2783 4542 a 2793 4615 4 183 v 521 4706 4 92 v 1748 4706 V 2176 4706 V 2793 4706 V 523 4710 2272 4 v Black Black 1922 5173 a FD(13)p Black eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.14) cvn /DEST pdfmark end 515 125 a Black Black 523 365 a SDict begin H.S end 523 365 a 523 365 a SDict begin 12 H.A end 523 365 a 523 365 a SDict begin [/View [/XYZ H.V]/Dest (section.A.2) cvn /DEST pdfmark end 523 365 a 83 x Fx(B)112 b(Pro)m(ving)37 b(the)g(main)i (Theorem)523 553 y SDict begin H.S end 523 553 a 523 553 a SDict begin 12 H.A end 523 553 a 523 553 a SDict begin [/View [/XYZ H.V]/Dest (theorem.dummy.3) cvn /DEST pdfmark end 523 553 a Black 80 x Fr(Theorem)30 b(2.)p Black 41 w Fp(F)-6 b(or)36 b(any)g(set)f(of)h(items)g Fv(S)i FD(=)33 b Fs(f)p Fv(x)2113 645 y Ft(i)2141 633 y Fs(g)2183 603 y Ft(n)2183 655 y(i)p FC(=1)2294 633 y Fp(,)j FD(0)d Fv(<)g(\015)38 b Fs(\024)33 b Fv(\015)2755 603 y Fq(\003)2826 633 y FD(=)g(0)p Fv(:)p FD(003176)f Fp(and)523 733 y(any)e(p)l(ositive) h(inte)l(ger)f Fv(k)c Fs(\024)d FD(1)p Fv(=\015)5 b Fp(:)1011 937 y SDict begin H.S end 1011 937 a 1011 937 a SDict begin 12 H.A end 1011 937 a 1011 937 a SDict begin [/View [/XYZ H.V]/Dest (equation.\(restate\040dummy.3\)1.4.4) cvn /DEST pdfmark end 1011 937 a FD(E)1068 952 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1])1357 937 y FD([)p Fv(v)s FD(\()p Fv(C)1514 949 y Ft(k)q(;\015)1614 937 y FD(\()p Fv(S;)14 b(\022)r FD(\)\)])25 b Fs(\025)2082 880 y FD(1)p 1984 917 237 4 v 1984 994 a(1)18 b(+)g Fv(\015)5 b(k)2231 937 y FD(\(1)18 b Fs(\000)g Fv(\014)t FD(\()p Fv(\015)5 b(;)14 b(k)s FD(\)\))p Fv(T)2733 949 y Ft(k)2774 937 y FD(\()p Fv(S)5 b FD(\))p Fv(;)382 b FD(\(4\))523 1161 y Fp(wher)l(e)31 b Fv(\014)t FD(\()p Fv(\015)5 b(;)14 b(k)s FD(\))25 b(=)f(7)p Fv(:)p FD(4)1225 1090 y Fu(p)p 1307 1090 326 4 v 1307 1161 a Fv(\015)18 b FD(ln)o(\(1)p Fv(=\015)5 b FD(\))19 b(+)1774 1128 y FC(5)p 1745 1142 92 4 v 1745 1151 a Fq(p)p 1800 1151 37 3 v 49 x Ft(k)1846 1161 y Fv(;)31 b Fp(and)g(the)f(exp)l(e)l(ctation)h(is)g(taken)f(oven)h (al)t(l)h(uni-)523 1276 y(form)43 b(mappings)h(of)f Fv(S)k Fp(to)42 b(the)h(interval)g FD([0)p Fv(;)14 b FD(1\))p Fp(.)42 b(\(Note)g(that)g(the)h(right)g(hand)g(side)g(of)523 1376 y(Equation)30 b(\()910 1392 y SDict begin H.S end 910 1392 a Black -16 x Fp(4)p Black 953 1322 a SDict begin H.R end 953 1322 a 953 1376 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.4.4) cvn H.B /ANN pdfmark end 953 1376 a Fp(\))f(is)i(ne)l(gative)f(for)h Fv(\015)1607 1346 y Fq(\003)1667 1376 y Fv(<)23 b(\015)28 b Fs(\024)22 b FD(0)p Fv(:)p FD(5)p Fp(.\))p Black 523 1532 a(Pr)l(o)l(of.)p Black 43 w FD(The)34 b(pro)r(of)e(is)g(via)g(induction)g(on)h Fv(k)s FD(.)g(F)-7 b(or)33 b(the)h(base)e(case)h(of)g(the)h(induction)d (w)n(e)523 1652 y(note)k(that)h(for)f(an)n(y)g Fv(k)k(<)d FD(25)f(the)h(theorem)e(holds)g(since)2423 1560 y Fu(\020)2473 1652 y FD(1)18 b Fs(\000)2655 1620 y FC(5)p 2626 1634 92 4 v 2626 1642 a Fq(p)p 2680 1642 37 3 v 2680 1692 a Ft(k)2727 1560 y Fu(\021)2813 1652 y Fs(\024)36 b FD(0.)f(W)-7 b(e)36 b(hereb)n(y)523 1777 y(assume)26 b(that)i(the)g(theorem)e(holds) g(for)h(an)n(y)g Fv(k)2003 1747 y Fq(0)2049 1777 y Fs(\024)c Fv(k)s FD(,)k(and)h(pro)n(v)n(e)e(the)i(statemen)n(t)e(for)h Fv(k)s FD(.)648 1877 y(The)34 b(pro)r(of)f(is)g(presen)n(ted)h(top-do)n (wn.)f(W)-7 b(e)35 b(refer)e(to)h(Lemmata)2746 1877 y SDict begin H.S end 2746 1877 a Black FD(1)p Black 2788 1823 a SDict begin H.R end 2788 1823 a 2788 1877 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.1) cvn H.B /ANN pdfmark end 2788 1877 a FD(,)2845 1877 y SDict begin H.S end 2845 1877 a Black FD(2)p Black 2887 1823 a SDict begin H.R end 2887 1823 a 2887 1877 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.2) cvn H.B /ANN pdfmark end 2887 1877 a 34 w FD(and)3089 1877 y SDict begin H.S end 3089 1877 a Black FD(5)p Black 3130 1823 a SDict begin H.R end 3130 1823 a 3130 1877 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 3130 1877 a FD(,)h(whose)523 1976 y(statemen)n(t)27 b(and)g(pro)r(of)g(app)r (ear)g(subsequen)n(tly)-7 b(.)648 2076 y(Let)24 b Fv(C)858 2046 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1060 2076 y FD(\()p Fv(X)r(;)14 b(\022)r FD(\))24 b(and)g Fv(C)1520 2046 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1722 2076 y FD(\()p Fv(X)r(;)14 b(\022)r FD(\))24 b(b)r(e)g(the)g(subsets)g(of)g Fv(X)30 b FD(c)n(hosen)23 b(b)n(y)g(algorithm)523 2175 y Fv(C)582 2187 y Ft(k)q(;\015)706 2175 y FD(during)g(the)j(time)d(in)n (terv)-5 b(als)22 b([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))23 b(and)h([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\)\))24 b(resp)r(ectiv)n(ely)e (when)i(applied)f(to)523 2275 y Fv(X)r(;)14 b(\022)r FD(.)648 2375 y(F)-7 b(or)27 b(the)h(induction)e(step)h(w)n(e)g(use)h (the)g(fact)g(that)811 2539 y SDict begin H.S end 811 2539 a 811 2539 a SDict begin 12 H.A end 811 2539 a 811 2539 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.9) cvn /DEST pdfmark end 811 2539 a FD(E)868 2554 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1164 2539 y FD([)p Fv(v)s FD(\()q Fv(C)1322 2551 y Ft(k)q(;\015)1421 2539 y FD(\()p Fv(S;)14 b(\022)r FD(\)\))q(])834 2698 y(=)23 b(E)979 2713 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1275 2605 y Fu(h)1314 2698 y Fv(v)1357 2605 y Fu(\020)1407 2698 y Fv(C)1472 2654 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1466 2723 y Ft(k)q(;\015)1674 2698 y FD(\()p Fv(S;)14 b(\022)r FD(\))1867 2605 y Fu(\021)q(i)1976 2698 y FD(+)k(E)2116 2713 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2412 2605 y Fu(h)2451 2698 y Fv(v)2494 2605 y Fu(\020)2544 2698 y Fv(C)2609 2654 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2603 2723 y Ft(k)q(;\015)2811 2698 y FD(\()p Fv(S;)c(\022)r FD(\))3004 2605 y Fu(\021)q(i)p Black 3299 2698 a FD(\(9\))p Black 523 2887 a(W)-7 b(e)32 b(giv)n(e)e(a)h(lo)n(w)n(er)e(b)r(ound)j(on)f(the)h(\014rst)g(term)e (in)h(\()2176 2887 y SDict begin H.S end 2176 2887 a Black FD(9)p Black 2218 2834 a SDict begin H.R end 2218 2834 a 2218 2887 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.9) cvn H.B /ANN pdfmark end 2218 2887 a FD(\))h(using)e(the)i (induction)e(h)n(yp)r(othesis)523 2987 y(\(after)j(an)f(appropriate)e (transformation\))f(and)j(w)n(e)g(directly)f(lo)n(w)n(er)f(b)r(ound)j (the)g(second)523 3086 y(term)27 b(using)f(Lemma)1235 3086 y SDict begin H.S end 1235 3086 a Black FD(5)p Black 1277 3033 a SDict begin H.R end 1277 3033 a 1277 3086 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 1277 3086 a FD(.)648 3186 y(Since)f(Lemma)1158 3186 y SDict begin H.S end 1158 3186 a Black FD(5)p Black 1200 3133 a SDict begin H.R end 1200 3133 a 1200 3186 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 1200 3186 a 26 w FD(requires)f(that)j(the)g(size)e Fv(n)h FD(of)h Fv(S)k FD(is)25 b(su\016cien)n(tly)g(large)f(relativ)n (e)f(to)j Fv(k)523 3286 y FD(\(whic)n(h)k(ma)n(y)f(not)h(b)r(e)h(the)g (case\))f(w)n(e)g(in)n(tro)r(duce)f(a)h(mo)r(di\014cation)e(of)i Fv(C)2808 3298 y Ft(k)q(;\015)2908 3286 y FD(,)g(Algorithm)3363 3286 y SDict begin H.S end 3363 3286 a Black FD(2)p Black 3405 3232 a SDict begin H.R end 3405 3232 a 3405 3286 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (algocfline.2) cvn H.B /ANN pdfmark end 3405 3286 a 523 3385 a FD(|)e(denoted)f(b)n(y)h Fv(C)1128 3355 y Fq(\003)1122 3409 y Ft(k)q(;\015)1221 3385 y FD(:)1299 3385 y SDict begin H.S end 1299 3385 a 1299 3385 a SDict begin 12 H.A end 1299 3385 a 1299 3385 a SDict begin [/View [/XYZ H.V]/Dest (algocfline.2) cvn /DEST pdfmark end 1299 3385 a 648 3485 a FD(By)f(Lemma)1074 3485 y SDict begin H.S end 1074 3485 a Black FD(1)p Black 1116 3431 a SDict begin H.R end 1116 3431 a 1116 3485 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.1) cvn H.B /ANN pdfmark end 1116 3485 a 27 w FD(w)n(e)h(ha)n(v)n(e)e(that)1020 3650 y SDict begin H.S end 1020 3650 a 1020 3650 a SDict begin 12 H.A end 1020 3650 a 1020 3650 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.10) cvn /DEST pdfmark end 1020 3650 a FD(E)1077 3665 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1374 3650 y FD([)p Fv(v)s FD(\()p Fv(C)1531 3662 y Ft(k)q(;\015)1631 3650 y FD(\()p Fv(S;)14 b(\022)r FD(\)\))q(])23 b Fs(\025)g FD(E)2048 3665 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2344 3582 y Fu(\002)2379 3650 y Fv(v)2422 3582 y Fu(\000)2460 3650 y Fv(C)2525 3615 y Fq(\003)2519 3670 y Ft(k)q(;\015)2618 3650 y FD(\()p Fv(S;)14 b(\022)r FD(\))2811 3582 y Fu(\001)q(\003)2885 3650 y Fv(:)349 b FD(\(10\))523 3814 y(By)27 b(de\014nition)f(of)i Fv(C)1182 3784 y Fq(\003)1176 3838 y Ft(k)q(;\015)1303 3814 y FD(ab)r(o)n(v)n(e)e (w)n(e)h(ha)n(v)n(e)g(that)598 3993 y SDict begin H.S end 598 3993 a 598 3993 a SDict begin 12 H.A end 598 3993 a 598 3993 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.11) cvn /DEST pdfmark end 598 3993 a FD(E)655 4008 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))951 3926 y Fu(\002)986 3993 y Fv(v)1029 3926 y Fu(\000)1067 3993 y Fv(C)1132 3959 y Fq(\003)1126 4013 y Ft(k)q(;\015)1225 3993 y FD(\()p Fv(S;)14 b(\022)r FD(\))1418 3926 y Fu(\001)q(\003)621 4153 y FD(=)23 b(E)766 4168 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1062 4061 y Fu(h)1101 4153 y Fv(v)1144 4061 y Fu(\020)1194 4153 y Fv(C)1259 4110 y Fq(\003)p FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1253 4179 y Ft(k)q(;\015)1495 4153 y FD(\()p Fv(S;)14 b(\022)r FD(\))1688 4061 y Fu(\021)q(i)1797 4153 y FD(+)k(E)1937 4168 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2233 4061 y Fu(h)2272 4153 y Fv(v)2315 4061 y Fu(\020)2365 4153 y Fv(C)2430 4110 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2424 4179 y Ft(k)q(;\015)2666 4153 y FD(\()p Fv(S;)c(\022)r FD(\))2859 4061 y Fu(\021)q(i)621 4336 y FD(=)23 b(E)766 4351 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1062 4244 y Fu(h)1101 4336 y Fv(v)1144 4244 y Fu(\020)1194 4336 y Fv(C)1259 4293 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1253 4361 y Ft(k)q(;\015)1461 4336 y FD(\()p Fv(S;)14 b(\022)r FD(\))1654 4244 y Fu(\021)q(i)1763 4336 y FD(+)k(E)1903 4351 y Ft(\022)1937 4335 y Fd(0)1958 4351 y FC(:\()p Ft(S)s Fq([)p Ft(D)r FC(\))p Fq(7!)p FC([0)p Ft(;)p FC(1\))2374 4244 y Fu(h)2414 4336 y Fv(v)2457 4244 y Fu(\020)2506 4336 y Fv(C)2571 4293 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2565 4361 y Ft(k)q(;\015)2774 4336 y FD(\()p Fv(S)23 b Fs([)c Fv(D)r(;)14 b(\022)3103 4302 y Fq(0)3126 4336 y FD(\))3158 4244 y Fu(\021i)p Black 3257 4336 a FD(\(11\))p Black 648 4526 a(The)21 b(mapping)f Fv(\022)1189 4496 y Fq(0)1234 4526 y FD(in)h(Equation)f(\()1710 4526 y SDict begin H.S end 1710 4526 a Black FD(11)p Black 1793 4472 a SDict begin H.R end 1793 4472 a 1793 4526 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 1793 4526 a FD(\),)i(with)f(domain)e Fv(S)11 b Fs([)c Fv(D)r FD(,)22 b(com)n(bines)d Fv(\022)26 b FD(:)d Fv(S)28 b Fs(7!)23 b FD([0)p Fv(;)14 b FD(1\))523 4625 y(and)27 b Fv(\022)723 4637 y Ft(D)807 4625 y FD(:)c Fv(D)i Fs(7!)e FD([0)p Fv(;)14 b FD(1\),)27 b(this)g(is)f(w)n(ell)g (de\014ned)i(b)r(ecause)f Fv(S)32 b FD(and)c Fv(D)h FD(are)e(disjoin)n (t.)648 4725 y(W)-7 b(e)23 b(giv)n(e)e(a)i(lo)n(w)n(er)e(b)r(ound)i (for)g(the)h(\014rst)f(term)f(in)g(Equation)g(\()2583 4725 y SDict begin H.S end 2583 4725 a Black FD(11)p Black 2666 4672 a SDict begin H.R end 2666 4672 a 2666 4725 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 2666 4725 a FD(\))h(using)f(the)i(inductiv)n(e)523 4825 y(h)n(yp)r(othesis.)18 b(Using)g(Lemma)1460 4825 y SDict begin H.S end 1460 4825 a Black FD(5)p Black 1502 4771 a SDict begin H.R end 1502 4771 a 1502 4825 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 1502 4825 a 19 w FD(w)n(e)h(deriv)n(e)e(a)i(lo)n(w)n(er)e (b)r(ound)j(for)e(the)i(2nd)f(term)f(in)h(Equation)523 4924 y(\()555 4924 y SDict begin H.S end 555 4924 a Black FD(11)p Black 638 4871 a SDict begin H.R end 638 4871 a 638 4924 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 638 4924 a FD(\).)p Black 1922 5173 a(14)p Black eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end 515 125 a Black Black Black 523 397 2882 7 v Black 523 468 a Fz(ALGORITHM)30 b(2:)e Fy(C)1279 437 y Fi(\003)1274 484 y Fj(k)q(;\015)p Black 523 517 V Black 523 609 a SDict begin H.S end 523 609 a 523 609 a SDict begin 11 H.A end 523 609 a 523 609 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.1) cvn /DEST pdfmark end 523 609 a -79 w Fg(1)41 b FB(During)25 b(the)h(time)f(in)n(terv)l(al)g([0)p Fy(;)14 b FB(1)p Fy(=)p FB(2\),)27 b Fy(C)1713 577 y Fi(\003)1708 624 y Fj(k)q(;\015)1826 609 y FB(em)n(ulates)e Fy(C)2199 618 y Fj(k)q(;\015)2292 609 y FB(,)h Fc(i.e.)p FB(,)h Fy(C)2544 571 y Fi(\003)p Fn([0)p Fj(;)p Fn(1)p Fj(=)p Fn(2\))2539 630 y Fj(k)q(;\015)2764 609 y FB(\()p Fy(S)t FB(\))21 b(=)g Fy(C)3037 571 y Fn([0)p Fj(;)p Fn(1)p Fj(=)p Fn(2\))3032 630 y Fj(k)q(;\015)3225 609 y FB(\()p Fy(S)t FB(\).)523 712 y SDict begin H.S end 523 712 a 523 712 a SDict begin 11 H.A end 523 712 a 523 712 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.2) cvn /DEST pdfmark end 523 712 a -79 w Fg(2)41 b Fy(D)24 b Fk( )h FB(a)h(collection)i(of)e(3)p Fk(d)q Fy(k)r(=)p FB(2)q Fk(e)f FB(distinguishable)i(dumm)n(y)22 b(items)j(with)h(zero)h(v)l (alue.)523 804 y SDict begin H.S end 523 804 a 523 804 a SDict begin 11 H.A end 523 804 a 523 804 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.3) cvn /DEST pdfmark end 523 804 a -79 w Fg(3)41 b Fy(\022)559 812 y Fj(D)636 804 y Fk( )25 b FB(random)g(uniform)g(mapping)g Fy(D)f Fk(7!)d FB([0)p Fy(;)14 b FB(1\))523 895 y SDict begin H.S end 523 895 a 523 895 a SDict begin 11 H.A end 523 895 a 523 895 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.4) cvn /DEST pdfmark end 523 895 a -79 w Fg(4)41 b Fy(U)30 b Fk( )21 b(f)p Fy(x)g Fk(2)h Fy(S)j Fk(j)d Fy(\022)r FB(\()p Fy(x)p FB(\))e Fy(<)h FB(1)p Fy(=)p FB(2)p Fk(g)e([)e(f)p Fy(d)22 b Fk(2)f Fy(D)j Fk(j)e Fy(\022)1814 903 y Fj(D)1869 895 y FB(\()p Fy(d)p FB(\))f Fy(<)g FB(1)p Fy(=)p FB(2)p Fk(g)p Fy(:)p Black 523 986 a FA(/*)p Black 40 w(set)40 b(the)g(threshold)h Fy(T)1403 955 y Fi(\003)3326 986 y FA(*/)523 1078 y SDict begin H.S end 523 1078 a 523 1078 a SDict begin 11 H.A end 523 1078 a 523 1078 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.5) cvn /DEST pdfmark end 523 1078 a -79 w Fg(5)g Fy(\034)30 b FB(=)21 b Fk(d)p Fy(k)r(=)p FB(2)q Fk(e)523 1169 y SDict begin H.S end 523 1169 a 523 1169 a SDict begin 11 H.A end 523 1169 a 523 1169 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.6) cvn /DEST pdfmark end 523 1169 a -79 w Fg(6)41 b Fz(if)34 b Fk(j)p Fy(U)8 b Fk(j)22 b(\025)f Fy(\034)34 b Fz(then)p 565 1470 4 274 v 651 1260 a SDict begin H.S end 651 1260 a 651 1260 a SDict begin 11 H.A end 651 1260 a 651 1260 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.7) cvn /DEST pdfmark end 651 1260 a -207 w Fg(7)169 b Fy(T)707 1229 y Fi(\003)764 1260 y Fk( )25 b FB(the)g Fy(\034)34 b FB(largest)27 b(v)l(alue)f(amongst)f Fy(U)p Black 651 1352 a FA(/*)p Black 40 w(Note)40 b(that)g(if)g(the)f(number)i(of)f(items)g(from)g Fy(S)k FA(mapped)c(to)g(the)g(interval)769 1443 y FB([0)p Fy(;)13 b FB(1)p Fy(=)p FB(2\))41 b FA(is)e Fk(\025)21 b Fy(\034)9 b FA(,)39 b(then)h Fy(C)1622 1411 y Fi(\003)1617 1458 y Fj(k)q(;\015)1749 1443 y FA(sets)g(the)g(same)g(threshold)i(as)d Fy(C)2863 1452 y Fj(k)q(;\015)3323 1443 y FA(*/)523 1543 y SDict begin H.S end 523 1543 a 523 1543 a SDict begin 11 H.A end 523 1543 a 523 1543 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.8) cvn /DEST pdfmark end 523 1543 a -79 w Fg(8)i Fz(else)p 565 1665 4 95 v 651 1634 a SDict begin H.S end 651 1634 a 651 1634 a SDict begin 11 H.A end 651 1634 a 651 1634 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.9) cvn /DEST pdfmark end 651 1634 a -207 w Fg(9)169 b Fy(T)707 1602 y Fi(\003)764 1634 y Fk( )25 b FB(absolute)h(zero)p Black 920 w FA(//)p Black 39 w(smaller)41 b(than)f(any)g(other)h(item)p 569 1665 39 4 v Black 523 1757 a(/*)p Black 40 w(the)f(arrival)g(time)h(of)e(the)h(next)g(item)g (will)h(be)e(after)i(time)f FB(1)p Fy(=)p FB(2)492 b FA(*/)523 1848 y SDict begin H.S end 523 1848 a 523 1848 a SDict begin 11 H.A end 523 1848 a 523 1848 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.10) cvn /DEST pdfmark end 523 1848 a -117 w Fg(10)41 b Fy(x)21 b Fk( )k FB(next)g(item)g(from)h Fy(S)21 b Fk([)c Fy(D)p Black 568 w FA(//)p Black 40 w(If)40 b(no)g(further)g(items)h(arrive,)g Fy(x)20 b Fk( )i(;)523 1940 y SDict begin H.S end 523 1940 a 523 1940 a SDict begin 11 H.A end 523 1940 a 523 1940 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.11) cvn /DEST pdfmark end 523 1940 a -117 w Fg(11)41 b Fz(while)24 b Fy(x)d Fk(6)p FB(=)g Fk(;)56 b Ff(AND)e(no)28 b(mor)l(e)g(than)h Fy(k)19 b Fk(\000)e FB(1)28 b Ff(items)f(have)i(alr)l(e)l(ady)f(b)l(e)l (en)h(ac)l(c)l(epte)l(d)36 b Fz(do)p 565 2368 4 401 v 651 2031 a SDict begin H.S end 651 2031 a 651 2031 a SDict begin 11 H.A end 651 2031 a 651 2031 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.12) cvn /DEST pdfmark end 651 2031 a -245 w Fg(12)169 b Fz(if)34 b Fy(x)20 b(>)h(T)938 1999 y Fi(\003)1029 2031 y Ff(AND)55 b(\()p Fy(\022)r FB(\()p Fy(x)p FB(\))21 b Fk(\025)g Fy(\022)r FB(\()p Fy(x)1643 1999 y Fi(0)1665 2031 y FB(\))16 b(+)h Fy(\015)32 b Ff(wher)l(e)c Fy(x)2120 1999 y Fi(0)2170 2031 y Ff(is)f(the)h(last)g (item)g(ac)l(c)l(epte)l(d)i(by)e Fy(C)3157 1999 y Fi(\003)3152 2047 y Fj(k)q(;\015)651 2122 y SDict begin H.S end 651 2122 a 651 2122 a SDict begin 11 H.A end 651 2122 a 651 2122 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.13) cvn /DEST pdfmark end 651 2122 a -245 w Fg(13)510 b Ff(OR)55 b(no)27 b(items)h(have)h(b)l(e)l(en)f(pr)l(eviously)h(ac)l(c)l(epte)l (d\))i Fz(then)p 692 2244 4 95 v 779 2214 a SDict begin H.S end 779 2214 a 779 2214 a SDict begin 11 H.A end 779 2214 a 779 2214 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.14) cvn /DEST pdfmark end 779 2214 a -373 w Fg(14)297 b Fy(C)839 2182 y Fi(\003)834 2229 y Fj(k)q(;\015)952 2214 y FB(accepts)26 b Fy(x)p 696 2244 39 4 v 651 2337 a SDict begin H.S end 651 2337 a 651 2337 a SDict begin 11 H.A end 651 2337 a 651 2337 a SDict begin [/View [/XYZ H.V]/Dest (AlgoLine.2.15) cvn /DEST pdfmark end 651 2337 a -245 w Fg(15)169 b Fy(x)21 b Fk( )k FB(next)g(item)g(from)g Fy(S)d Fk([)17 b Fy(D)p Black 437 w FA(//)p Black 40 w(If)40 b(no)f(further)i(items)f(arrive,)h Fy(x)21 b Fk( )g(;)p 569 2368 39 4 v 523 2396 a SDict begin H.S end 523 2396 a 523 2396 a SDict begin 11 H.A end 523 2396 a 523 2396 a SDict begin [/View [/XYZ H.V]/Dest (algocf.2) cvn /DEST pdfmark end 523 2396 a Black 523 2420 2882 7 v Black 648 2925 a FD(T)-7 b(o)32 b(simplify)e(the)j(notation)f (hereinafter)f(w)n(e)h(abbreviate)f(E)2602 2940 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2899 2925 y FD([)p Fv(v)s FD(\()p Fv(C)3056 2937 y Ft(k)q(;\015)3156 2925 y FD(\()p Fv(S;)14 b(\022)r FD(\)\)])523 3025 y(as)27 b(E[)p Fv(C)764 3037 y Ft(k)q(;\015)863 3025 y FD(\()p Fv(S)5 b FD(\)].)648 3229 y(W)-7 b(e)30 b(\014rst)g(giv)n(e)e(a)i(lo)n (w)n(er)d(b)r(ound)j(on)g(the)h(\014rst)e(term)g(in)h(Equation)e(\() 2820 3229 y SDict begin H.S end 2820 3229 a Black FD(11)p Black 2903 3175 a SDict begin H.R end 2903 3175 a 2903 3229 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 2903 3229 a FD(\).)j(Giv)n(en)d(a)i(set)523 3328 y(of)25 b(items)d Fv(S)5 b FD(,)25 b(\014x)f(the)h(set)g(of)f (items)f Fv(S)1684 3298 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1909 3328 y Fs(\022)f Fv(S)30 b FD(arriving)20 b(in)k([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\).)23 b(The)i(arriv)-5 b(al)21 b(times)523 3428 y(of)31 b Fv(S)677 3398 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))910 3428 y FD(are)g(uniform)e(in)h([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\).)30 b(Therefore)g(conditioned)f(on)i Fv(S)2787 3398 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))3020 3428 y FD(arriving)d(in)523 3527 y([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\),)26 b(the)i(exp)r(ected)g(pro\014t)f(of)h Fv(C)1701 3539 y Ft(k)q(;\015)1828 3527 y FD(from)e Fv(S)2080 3497 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2309 3527 y FD(equals)1547 3947 y(E)1604 3854 y Fu(h)1643 3947 y Fv(C)1702 3962 y Fq(b)p Ft(k)q(=)p FC(2)q Fq(c)q Ft(;)p FC(2)p Ft(\015)1962 3854 y Fu(\020)2012 3947 y Fv(S)2068 3912 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2269 3854 y Fu(\021i)2358 3947 y Fv(:)648 4462 y FD(By)h(induction)f(w)n(e)h (obtain)601 4888 y(E)658 4796 y Fu(h)697 4888 y Fv(C)756 4903 y Fq(b)q Ft(k)q(=)p FC(2)q Fq(c)p Ft(;)p FC(2)p Ft(\015)1016 4796 y Fu(\020)1066 4888 y Fv(S)1122 4854 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1324 4796 y Fu(\021)o(i)1436 4888 y Fs(\025)1730 4832 y FD(1)p 1533 4869 435 4 v 1533 4945 a(1)18 b(+)g(2)p Fv(\015)5 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)1978 4821 y Fu(\000)2016 4888 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)p Fv(k)s(=)p FD(2)o Fs(c)o FD(\))2603 4821 y Fu(\001)2660 4888 y Fs(\001)19 b Fv(T)2751 4903 y Fq(b)p Ft(k)q(=)p FC(2)q Fq(c)2919 4796 y Fu(\020)2969 4888 y Fv(S)3025 4854 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))3226 4796 y Fu(\021)3304 4888 y Fv(:)p Black 1922 5173 a FD(15)p Black eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(It)26 b(therefore)f(follo)n(ws)e(that)k(the) f(exp)r(ected)h(pro\014t)f(of)g(algorithm)c Fv(C)2773 460 y Ft(k)q(;\015)2898 448 y FD(from)j(elemen)n(ts)523 548 y(arriving)f(in)j([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))26 b(\(without)h(an)n(y)g(conditioning)d(on)j Fv(S)2362 518 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2564 548 y FD(\),)h(is)799 728 y SDict begin H.S end 799 728 a 799 728 a SDict begin 12 H.A end 799 728 a 799 728 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.12) cvn /DEST pdfmark end 799 728 a FD(E)856 636 y Fu(h)895 728 y Fv(C)960 685 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))954 753 y Ft(k)q(;\015)1162 728 y FD(\()p Fv(S)5 b FD(\))1282 636 y Fu(i)823 910 y FD(=)22 b(E)967 929 y Ft(S)1011 913 y Fb([0)p Fo(;)p Fb(1)p Fo(=)p Fb(2\))1189 929 y Fq(\022)p Ft(S)1289 818 y Fu(h)1328 910 y FD(E)1385 818 y Fu(h)1423 910 y Fv(C)1482 925 y Fq(b)q Ft(k)q(=)p FC(2)q Fq(c)p Ft(;)p FC(2)p Ft(\015)1743 818 y Fu(\020)1792 910 y Fv(S)1848 876 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2050 818 y Fu(\021ii)823 1118 y Fs(\025)g FD(E)967 1137 y Ft(S)1011 1120 y Fb([0)p Fo(;)p Fb(1)p Fo(=)p Fb(2\))1189 1137 y Fq(\022)p Ft(S)1289 1001 y Fu(\024)1539 1062 y FD(1)p 1342 1099 435 4 v 1342 1175 a(1)c(+)g(2)p Fv(\015)5 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)1787 1051 y Fu(\000)1825 1118 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)p Fv(k)s(=)p FD(2)n Fs(c)p FD(\))2412 1051 y Fu(\001)2469 1118 y Fs(\001)19 b Fv(T)2560 1133 y Fq(b)p Ft(k)q(=)p FC(2)p Fq(c)2728 1026 y Fu(\020)2778 1118 y Fv(S)2834 1084 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))3035 1026 y Fu(\021)3085 1001 y(\025)823 1340 y FD(=)1117 1283 y(1)p 920 1321 V 920 1397 a(1)f(+)g(2)p Fv(\015)5 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)1365 1272 y Fu(\000)1403 1340 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)p Fv(k)s(=)p FD(2)o Fs(c)o FD(\))1990 1272 y Fu(\001)2047 1340 y Fs(\001)19 b FD(E)2146 1359 y Ft(S)2190 1342 y Fb([0)p Fo(;)p Fb(1)p Fo(=)p Fb(2\))2367 1359 y Fq(\022)p Ft(S)2467 1247 y Fu(h)2506 1340 y Fv(T)2555 1355 y Fq(b)p Ft(k)q(=)p FC(2)q Fq(c)2723 1247 y Fu(\020)2773 1340 y Fv(S)2829 1305 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))3031 1247 y Fu(\021)o(i)823 1560 y Fs(\025)1018 1504 y FD(1)p 920 1541 237 4 v 920 1617 a(1)f(+)g Fv(\015)5 b(k)1167 1493 y Fu(\000)1205 1560 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)p FD(\))1792 1493 y Fu(\001)1849 1560 y Fs(\001)k FD(E)1947 1579 y Ft(S)1991 1563 y Fb([0)p Fo(;)p Fb(1)p Fo(=)p Fb(2\))2169 1579 y Fq(\022)p Ft(S)2269 1468 y Fu(h)2308 1560 y Fv(T)2357 1575 y Fq(b)p Ft(k)q(=)p FC(2)p Fq(c)2525 1468 y Fu(\020)2575 1560 y Fv(S)2631 1526 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))2832 1468 y Fu(\021i)2949 1560 y Fv(:)p Black 285 w FD(\(12\))p Black 523 1771 a(The)26 b(set)g Fv(S)876 1740 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1104 1771 y FD(is)f(a)g(uniformly)e(random)h(subset)i(of) g Fv(S)5 b FD(,)26 b(therefore)f(applying)f(Lemma)3363 1771 y SDict begin H.S end 3363 1771 a Black FD(2)p Black 3405 1717 a SDict begin H.R end 3405 1717 a 3405 1771 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.2) cvn H.B /ANN pdfmark end 3405 1771 a 523 1870 a FD(w)n(e)j(get)1032 2015 y SDict begin H.S end 1032 2015 a 1032 2015 a SDict begin 12 H.A end 1032 2015 a 1032 2015 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.13) cvn /DEST pdfmark end 1032 2015 a FD(E)1089 2034 y Ft(S)1133 2018 y Fb([0)p Fo(;)p Fb(1)p Fo(=)p Fb(2\))1310 2034 y Fq(\022)p Ft(S)1410 1923 y Fu(h)1450 2015 y Fv(T)1499 2030 y Fq(b)p Ft(k)q(=)p FC(2)p Fq(c)1667 1923 y Fu(\020)1717 2015 y Fv(S)1773 1981 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1974 1923 y Fu(\021i)2086 2015 y Fs(\025)2174 1898 y Fu(\022)2235 2015 y FD(1)18 b Fs(\000)2445 1959 y FD(1)p 2388 1996 157 4 v 2388 2083 a(2)2430 2012 y Fs(p)p 2498 2012 46 4 v 2498 2083 a Fv(k)2554 1898 y Fu(\023)2625 1959 y Fv(T)2674 1971 y Ft(k)2715 1959 y FD(\()p Fv(S)5 b FD(\))p 2625 1996 210 4 v 2709 2072 a(2)2873 2015 y Fv(:)361 b FD(\(13\))523 2198 y(By)27 b(substituting)f(the)i(lo)n(w)n(er)d(b)r (ound)j(\()1763 2198 y SDict begin H.S end 1763 2198 a Black FD(13)p Black 1847 2144 a SDict begin H.R end 1847 2144 a 1847 2198 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.13) cvn H.B /ANN pdfmark end 1847 2198 a FD(\))g(in)n(to)e (Equation)g(\()2466 2198 y SDict begin H.S end 2466 2198 a Black FD(12)p Black 2549 2144 a SDict begin H.R end 2549 2144 a 2549 2198 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.12) cvn H.B /ANN pdfmark end 2549 2198 a FD(\))i(w)n(e)f(get)h(that)541 2377 y SDict begin H.S end 541 2377 a 541 2377 a SDict begin 12 H.A end 541 2377 a 541 2377 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.14) cvn /DEST pdfmark end 541 2377 a FD(E)598 2285 y Fu(h)637 2377 y Fv(C)702 2334 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))696 2402 y Ft(k)q(;\015)904 2377 y FD(\()p Fv(S)5 b FD(\))1024 2285 y Fu(i)626 2585 y Fs(\025)883 2529 y FD(1)p 785 2566 237 4 v 785 2642 a(1)18 b(+)g Fv(\015)5 b(k)1032 2518 y Fu(\000)1070 2585 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)p FD(\))1657 2518 y Fu(\001)1695 2468 y(\022)1757 2585 y FD(1)k Fs(\000)1967 2529 y FD(1)p 1910 2566 157 4 v 1910 2653 a(2)1952 2582 y Fs(p)p 2020 2582 46 4 v 2020 2653 a Fv(k)2076 2468 y Fu(\023)2147 2529 y Fv(T)2196 2541 y Ft(k)2236 2529 y FD(\()p Fv(S)5 b FD(\))p 2147 2566 210 4 v 2231 2642 a(2)626 2817 y Fs(\025)883 2761 y FD(1)p 785 2798 237 4 v 785 2874 a(1)18 b(+)g Fv(\015)5 b(k)1032 2700 y Fu(\022)1093 2817 y FD(1)18 b Fs(\000)g Fv(\014)t FD(\(2)p Fv(\015)5 b(;)14 b Fs(b)o Fv(k)s(=)p FD(2)o Fs(c)p FD(\))19 b Fs(\000)1849 2761 y FD(1)p 1792 2798 157 4 v 1792 2886 a(2)1834 2815 y Fs(p)p 1903 2815 46 4 v 71 x Fv(k)1958 2700 y Fu(\023)2029 2761 y Fv(T)2078 2773 y Ft(k)2119 2761 y FD(\()p Fv(S)5 b FD(\))p 2029 2798 210 4 v 2114 2874 a(2)565 3024 y FC(sub)p Ft(:)27 b(\014)626 3075 y FD(=)883 3018 y(1)p 785 3056 237 4 v 785 3132 a(1)18 b(+)g Fv(\015)5 b(k)1032 2933 y Fu( )1098 3075 y FD(1)17 b Fs(\000)h FD(7)p Fv(:)p FD(4)1347 3000 y Fu(p)p 1430 3000 474 4 v 75 x FD(2)p Fv(\015)g FD(ln)o(\(1)p Fv(=)p FD(\(2)p Fv(\015)5 b FD(\)\))17 b Fs(\000)2004 2933 y Fu( )2202 3018 y FD(5)p 2080 3056 286 4 v 2080 3072 a Fu(p)p 2163 3072 203 4 v 71 x Fs(b)p Fv(k)s(=)p FD(2)o Fs(c)2394 3075 y FD(+)2545 3018 y(1)p 2487 3056 157 4 v 2487 3143 a(2)2529 3072 y Fs(p)p 2598 3072 46 4 v 71 x Fv(k)2654 2933 y Fu(!)o(!)2795 3018 y Fv(T)2844 3030 y Ft(k)2885 3018 y FD(\()p Fv(S)5 b FD(\))p 2795 3056 210 4 v 2879 3132 a(2)626 3357 y Fs(\025)883 3301 y FD(1)p 785 3338 237 4 v 785 3414 a(1)18 b(+)g Fv(\015)5 b(k)1032 3215 y Fu( )1107 3301 y FD(1)p 1107 3338 42 4 v 1107 3414 a(2)1177 3357 y Fs(\000)1270 3232 y(p)p 1340 3232 V 1340 3301 a FD(2)p 1270 3338 111 4 v 1305 3414 a(2)1409 3357 y Fs(\001)19 b FD(7)p Fv(:)p FD(4)1558 3282 y Fu(p)p 1640 3282 326 4 v 1640 3357 a Fv(\015)g FD(ln)o(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\000)2067 3215 y Fu( )2143 3232 y Fs(p)p 2212 3232 42 4 v 69 x FD(2)p 2143 3338 111 4 v 2178 3414 a(2)2282 3357 y Fs(\001)2442 3301 y FD(5)p 2334 3338 259 4 v 2334 3354 a Fs(p)p 2403 3354 189 4 v 68 x Fv(k)k Fs(\000)d FD(1)2620 3357 y(+)2771 3301 y(1)p 2713 3338 157 4 v 2713 3425 a(4)2755 3354 y Fs(p)p 2824 3354 46 4 v 71 x Fv(k)2880 3215 y Fu(!)o(!)3011 3357 y Fv(T)3060 3369 y Ft(k)3100 3357 y FD(\()p Fv(S)5 b FD(\))p Black 37 w(\(14\))p Black 648 3586 a(where)27 b(the)h(last)e(inequalit)n(y)e(in)j(this)g(deriv)-5 b(ation)25 b(follo)n(ws)f(since)j Fs(b)p Fv(k)s(=)p FD(2)o Fs(c)22 b(\025)h FD(\()p Fv(k)f Fs(\000)c FD(1\))p Fv(=)p FD(2.)648 3686 y(No)n(w)24 b(w)n(e)g(giv)n(e)f(a)h(lo)n(w)n(er)e(b)r(ound)j(on)f (the)h(second)f(term)f(of)i(Equation)e(\()2846 3686 y SDict begin H.S end 2846 3686 a Black FD(11)p Black 2929 3633 a SDict begin H.R end 2929 3633 a 2929 3686 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 2929 3686 a FD(\).)i(By)f(Lemma)523 3786 y SDict begin H.S end 523 3786 a Black FD(5)p Black 565 3732 a SDict begin H.R end 565 3732 a 565 3786 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 565 3786 a 39 w FD(\(recall)37 b(that)j Fs(j)p Fv(S)23 b Fs([)c Fv(D)r Fs(j)43 b(\025)f FD(3)p Fs(d)p Fv(k)s(=)p FD(2)n Fs(e)p FD(\))e(w)n(e)f(obtain)g(that)g(the)h(exp)r (ected)g(pro\014t)f(of)h Fv(C)3306 3798 y Ft(k)q(;\015)523 3885 y FD(executed)28 b(on)f(input)g Fv(S)d Fs([)18 b Fv(D)30 b FD(during)c(the)i(time)e(in)n(terv)-5 b(al)25 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))26 b(is)h(at)g(least)738 4065 y SDict begin H.S end 738 4065 a 738 4065 a SDict begin 12 H.A end 738 4065 a 738 4065 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.15) cvn /DEST pdfmark end 738 4065 a FD(E)795 3973 y Fu(h)834 4065 y Fv(C)899 4022 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))893 4090 y Ft(k)q(;\015)1101 4065 y FD(\()p Fv(S)c Fs([)c Fv(D)r FD(\))1384 3973 y Fu(i)761 4298 y Fs(\025)957 4241 y FD(1)p 859 4278 237 4 v 859 4354 a(1)f(+)g Fv(\015)5 b(k)1106 4205 y Fu(\020)1155 4298 y FD(1)18 b Fs(\000)1298 4225 y(p)p 1367 4225 42 4 v 73 x FD(2)g Fs(\001)h Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)1891 4205 y Fu(\021)1939 4156 y( )2005 4298 y FD(1)g Fs(\000)2158 4240 y FD(2)2200 4169 y Fu(p)p 2282 4169 272 4 v 2282 4240 a FD(1)g(+)g(1)p Fv(=k)p 2158 4278 397 4 v 2298 4295 a Fs(p)p 2368 4295 46 4 v 2368 4366 a Fv(k)2583 4298 y Fs(\000)2678 4241 y FD(1)p 2675 4278 V 2675 4354 a Fv(k)2731 4156 y Fu(!)2807 4241 y Fv(T)2856 4253 y Ft(k)2896 4241 y FD(\()p Fv(S)24 b Fs([)19 b Fv(D)r FD(\))p 2807 4278 373 4 v 2973 4354 a(2)761 4580 y(=)957 4524 y(1)p 859 4561 237 4 v 859 4637 a(1)f(+)g Fv(\015)5 b(k)1106 4438 y Fu( )1181 4524 y FD(1)p 1181 4561 42 4 v 1181 4637 a(2)1251 4580 y Fs(\000)1334 4438 y Fu( )1410 4455 y Fs(p)p 1479 4455 V 69 x FD(2)p 1410 4561 111 4 v 1444 4637 a(2)1530 4580 y Fv(\013)p FD(\()p Fv(\015)g FD(\))20 b(+)1808 4524 y(9)p 1808 4561 42 4 v 1808 4637 a(4)1859 4580 y Fv(\015)1907 4438 y Fu(!)1991 4580 y Fs(\000)2074 4438 y Fu( )2149 4451 y(p)p 2232 4451 272 4 v 71 x FD(1)e(+)g(1)p Fv(=k)p 2149 4561 355 4 v 2269 4577 a Fs(p)p 2339 4577 46 4 v 2339 4648 a Fv(k)2533 4580 y FD(+)2649 4524 y(1)p 2626 4561 88 4 v 2626 4637 a(2)p Fv(k)2723 4438 y Fu(!!)2854 4580 y Fv(T)2903 4592 y Ft(k)2944 4580 y FD(\()p Fv(S)5 b FD(\))28 b Fv(;)p Black 142 w FD(\(15\))p Black 648 4825 a(where)e Fv(\013)p FD(\()p Fv(\015)5 b FD(\))24 b(=)e(3)1205 4754 y Fu(p)p 1288 4754 326 4 v 71 x Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))26 b(as)h(de\014ne)g(in)f(Theorem)2427 4825 y SDict begin H.S end 2427 4825 a Black FD(3)p Black 2468 4771 a SDict begin H.R end 2468 4771 a 2468 4825 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2468 4825 a FD(.)h(The)g(last)f(equalit)n(y)e(follo) n(ws)523 4924 y(since)i Fv(v)s FD(\()p Fv(D)r FD(\))e(=)f(0,)k (therefore)g Fv(T)1506 4936 y Ft(k)1546 4924 y FD(\()p Fv(S)d Fs([)19 b Fv(D)r FD(\))k(=)g Fv(T)1990 4936 y Ft(k)2030 4924 y FD(\()p Fv(S)5 b FD(\).)p Black 1922 5173 a(16)p Black eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Substituting)23 b(the)i(lo)n(w)n(er)d(b)r (ounds)j(\()1792 448 y SDict begin H.S end 1792 448 a Black FD(14)p Black 1875 395 a SDict begin H.R end 1875 395 a 1875 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.14) cvn H.B /ANN pdfmark end 1875 448 a FD(\))g(and)f(\()2122 448 y SDict begin H.S end 2122 448 a Black FD(15)p Black 2205 395 a SDict begin H.R end 2205 395 a 2205 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.15) cvn H.B /ANN pdfmark end 2205 448 a FD(\))h(in)n(to)f(Equation)e(\()2815 448 y SDict begin H.S end 2815 448 a Black FD(11)p Black 2898 395 a SDict begin H.R end 2898 395 a 2898 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.11) cvn H.B /ANN pdfmark end 2898 448 a FD(\))j(and)g(separat-)523 548 y(ing)h(terms)h(that)h(dep)r(end)g(on)f Fv(\015)32 b FD(from)27 b(those)g(that)h(dep)r(end)g(on)f Fv(k)k FD(w)n(e)c(get)542 851 y SDict begin H.S end 542 851 a 542 851 a SDict begin 12 H.A end 542 851 a 542 851 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.16) cvn /DEST pdfmark end 542 851 a FD(E)599 784 y Fu(\002)633 851 y Fv(C)698 817 y Fq(\003)692 872 y Ft(k)q(;\015)791 851 y FD(\()p Fv(S)5 b FD(\))911 784 y Fu(\003)969 851 y Fs(\025)1080 795 y Fv(T)1129 807 y Ft(k)1170 795 y FD(\()p Fv(S)g FD(\))p 1067 832 237 4 v 1067 908 a(1)18 b(+)g Fv(\015)5 b(k)1332 851 y Fs(\001)1373 709 y Fu( )1439 851 y FD(1)23 b Fs(\000)1591 709 y Fu( )1667 726 y Fs(p)p 1736 726 42 4 v 69 x FD(2)p 1667 832 111 4 v 1702 908 a(2)1788 851 y(7)p Fv(:)p FD(4)1895 776 y Fu(p)p 1977 776 326 4 v 1977 851 a Fv(\015)18 b FD(ln\(1)p Fv(=\015)5 b FD(\))17 b(+)2414 726 y Fs(p)p 2483 726 42 4 v 69 x FD(2)p 2414 832 111 4 v 2449 908 a(2)2535 851 y Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b(+)2812 795 y(9)p 2812 832 42 4 v 2812 908 a(4)2863 851 y Fv(\015)2911 709 y Fu(!)1504 1133 y Fs(\000)1591 991 y Fu( )1667 1008 y Fs(p)p 1736 1008 V 69 x FD(2)p 1667 1114 111 4 v 1702 1190 a(2)1806 1133 y Fs(\001)1966 1077 y FD(5)p 1858 1114 259 4 v 1858 1131 a Fs(p)p 1927 1131 189 4 v 67 x Fv(k)i Fs(\000)d FD(1)2144 1133 y(+)2295 1077 y(1)p 2237 1114 157 4 v 2237 1202 a(4)2279 1131 y Fs(p)p 2348 1131 46 4 v 71 x Fv(k)2422 1133 y FD(+)2515 1005 y Fu(p)p 2598 1005 272 4 v 71 x FD(1)g(+)g(1)p Fv(=k)p 2515 1114 355 4 v 2635 1131 a Fs(p)p 2704 1131 46 4 v 71 x Fv(k)2898 1133 y FD(+)3014 1077 y(1)p 2991 1114 88 4 v 2991 1190 a(2)p Fv(k)3089 991 y Fu(!)o(!)p Black 3257 1133 a FD(\(16\))p Black 648 1353 a(W)-7 b(e)28 b(observ)n(e)d(that)j(for)f(all)f Fv(\015)h Fs(\024)c Fv(\015)1715 1323 y Fq(\003)1776 1353 y FD(=)f(0)p Fv(:)p FD(003176)j(w)n(e)i(ha)n(v)n(e)1079 1434 y Fs(p)p 1148 1434 42 4 v 69 x FD(2)p 1079 1540 111 4 v 1114 1616 a(2)1200 1559 y(7)p Fv(:)p FD(4)1307 1484 y Fu(p)p 1389 1484 326 4 v 1389 1559 a Fv(\015)18 b FD(ln)o(\(1)p Fv(=\015)5 b FD(\))18 b(+)1826 1434 y Fs(p)p 1895 1434 42 4 v 69 x FD(2)p 1826 1540 111 4 v 1861 1616 a(2)1947 1559 y Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b(+)2224 1503 y(9)p 2224 1540 42 4 v 2224 1616 a(4)2275 1559 y Fv(\015)1092 1794 y FD(=)1180 1652 y Fu( )1256 1669 y Fs(p)p 1325 1669 V 69 x FD(2)p 1256 1775 111 4 v 1290 1851 a(2)1376 1794 y(7)p Fv(:)p FD(4)f(+)1594 1669 y Fs(p)p 1663 1669 42 4 v 69 x FD(2)p 1594 1775 111 4 v 1628 1851 a(2)1733 1794 y Fs(\001)h FD(3)e(+)1927 1738 y(9)p 1927 1775 42 4 v 1927 1851 a(4)1979 1683 y Fu(r)p 2062 1683 285 4 v 2180 1738 a Fv(\015)p 2072 1775 265 4 v 2072 1851 a FD(ln)o(\(1)p Fv(=\015)5 b FD(\))2346 1652 y Fu(!)2412 1719 y(p)p 2495 1719 326 4 v 75 x Fv(\015)18 b FD(ln\(1)p Fv(=\015)5 b FD(\))1092 2082 y Fs(\024)1180 1940 y Fu( )1256 1957 y Fs(p)p 1325 1957 42 4 v 68 x FD(2)p 1256 2062 111 4 v 1290 2139 a(2)1376 2082 y(7)p Fv(:)p FD(4)18 b(+)1594 1957 y Fs(p)p 1663 1957 42 4 v 68 x FD(2)p 1594 2062 111 4 v 1628 2139 a(2)1733 2082 y Fs(\001)h FD(3)e(+)1927 2025 y(9)p 1927 2062 42 4 v 1927 2139 a(4)1979 1938 y Fu(s)p 2062 1938 323 4 v 2180 2025 a Fv(\015)2228 2001 y Fq(\003)p 2072 2062 303 4 v 2072 2139 a FD(ln)o(\(1)p Fv(=\015)2305 2115 y Fq(\003)2342 2139 y FD(\))2384 1940 y Fu(!)2450 2007 y(p)p 2533 2007 326 4 v 75 x Fv(\015)i FD(ln)o(\(1)p Fv(=\015)5 b FD(\))1092 2300 y Fs(\031)23 b FD(7)p Fv(:)p FD(3407561)1539 2225 y Fu(p)p 1618 2225 V 1618 2300 a Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))22 b Fv(<)h(E)2116 2312 y FC(1)2153 2225 y Fu(p)p 2236 2225 V 75 x Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))27 b Fv(:)523 2450 y FD(Similarly)22 b(for)27 b Fv(k)f(>)d FD(25)j(w)n(e)i(ha)n(v)n(e)1028 2535 y Fs(p)p 1097 2535 42 4 v 68 x FD(2)p 1028 2640 111 4 v 1063 2716 a(2)1167 2659 y Fs(\001)1327 2603 y FD(5)p 1219 2640 259 4 v 1219 2657 a Fs(p)p 1288 2657 189 4 v 67 x Fv(k)21 b Fs(\000)d FD(1)1505 2659 y(+)1656 2603 y(1)p 1598 2640 157 4 v 1598 2728 a(4)1640 2657 y Fs(p)p 1709 2657 46 4 v 71 x Fv(k)1783 2659 y FD(+)1876 2531 y Fu(p)p 1959 2531 272 4 v 71 x FD(1)g(+)g(1)p Fv(=k)p 1876 2640 355 4 v 1996 2657 a Fs(p)p 2065 2657 46 4 v 71 x Fv(k)2259 2659 y FD(+)2375 2603 y(1)p 2352 2640 88 4 v 2352 2716 a(2)p Fv(k)1041 2915 y FD(=)1129 2773 y Fu( )1205 2790 y Fs(p)p 1274 2790 42 4 v 69 x FD(2)p 1205 2896 111 4 v 1240 2972 a(2)1492 2859 y(5)p 1336 2896 355 4 v 1336 2913 a Fu(p)p 1419 2913 272 4 v 71 x FD(1)g Fs(\000)g FD(1)p Fv(=k)1719 2915 y FD(+)1812 2859 y(1)p 1812 2896 42 4 v 1812 2972 a(4)1882 2915 y(+)1965 2840 y Fu(p)p 2048 2840 272 4 v 75 x FD(1)g(+)g(1)p Fv(=k)i FD(+)2489 2859 y(1)p 2431 2896 157 4 v 2431 2983 a(2)2473 2913 y Fs(p)p 2542 2913 46 4 v 70 x Fv(k)2598 2773 y Fu(!)2710 2859 y FD(1)p 2673 2896 116 4 v 2673 2913 a Fs(p)p 2742 2913 46 4 v 70 x Fv(k)1041 3197 y(<)1129 3055 y Fu( )1205 3073 y Fs(p)p 1274 3073 42 4 v 68 x FD(2)p 1205 3178 111 4 v 1240 3254 a(2)1511 3141 y(5)p 1336 3178 393 4 v 1336 3195 a Fu(p)p 1419 3195 310 4 v 71 x FD(1)e Fs(\000)g FD(1)p Fv(=)p FD(25)1756 3197 y(+)1849 3141 y(1)p 1849 3178 42 4 v 1849 3254 a(4)1919 3197 y(+)2002 3122 y Fu(p)p 2085 3122 310 4 v 75 x FD(1)g(+)g(1)p Fv(=)p FD(25)e(+)2582 3141 y(1)p 2505 3178 194 4 v 2505 3264 a(2)2547 3195 y Fs(p)p 2616 3195 84 4 v 69 x FD(25)2709 3055 y Fu(!)2822 3141 y FD(1)p 2785 3178 116 4 v 2785 3195 a Fs(p)p 2854 3195 46 4 v 71 x Fv(k)1041 3444 y Fs(\031)1139 3388 y FD(4)p Fv(:)p FD(9783)p 1139 3425 231 4 v 1197 3442 a Fs(p)p 1266 3442 46 4 v 70 x Fv(k)1403 3444 y(<)1537 3388 y FD(5)p 1500 3425 116 4 v 1500 3442 a Fs(p)p 1570 3442 46 4 v 1570 3512 a Fv(k)1653 3444 y(:)648 3642 y FD(Substituting)26 b(\()1155 3642 y SDict begin H.S end 1155 3642 a Black FD(16)p Black 1238 3588 a SDict begin H.R end 1238 3588 a 1238 3642 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.16) cvn H.B /ANN pdfmark end 1238 3642 a FD(\))i(in)n(to)f(\()1499 3642 y SDict begin H.S end 1499 3642 a Black FD(10)p Black 1582 3588 a SDict begin H.R end 1582 3588 a 1582 3642 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.10) cvn H.B /ANN pdfmark end 1582 3642 a FD(\))h(w)n(e)f(obtain)g(the)g(statemen)n(t)g(of)h (the)g(theorem.)241 b Fs(u)-55 b(t)523 3704 y SDict begin H.S end 523 3704 a 523 3704 a SDict begin 12 H.A end 523 3704 a 523 3704 a SDict begin [/View [/XYZ H.V]/Dest (lemma.1) cvn /DEST pdfmark end 523 3704 a Black 79 x Fr(Lemma)30 b(1.)p Black 40 w Fp(F)-6 b(or)30 b(any)g(set)g Fv(S)1020 3932 y FD(E)1077 3947 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1374 3932 y FD([)p Fv(v)s FD(\()p Fv(C)1531 3944 y Ft(k)q(;\015)1631 3932 y FD(\()p Fv(S;)14 b(\022)r FD(\)\))q(])23 b Fs(\025)g FD(E)2048 3947 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2344 3865 y Fu(\002)2379 3932 y Fv(v)2422 3865 y Fu(\000)2460 3932 y Fv(C)2525 3898 y Fq(\003)2519 3953 y Ft(k)q(;\015)2618 3932 y FD(\()p Fv(S;)14 b(\022)r FD(\))2811 3865 y Fu(\001)q(\003)2885 3932 y Fv(:)p Black 523 4082 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Since)26 b Fv(C)1064 4052 y Fq(\003)1058 4106 y Ft(k)q(;\015)1183 4082 y FD(runs)g(exactly)f(as)g Fv(C)1811 4094 y Ft(k)q(;\015)1937 4082 y FD(in)g(the)i(time)e(in)n(terv)-5 b(al)23 b([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))25 b(the)i(exp)r(ected)523 4182 y(v)-5 b(alue)34 b(of)h(b)r(oth)g(of)g(them)f(in)g(this)g(in)n (terv)-5 b(al)32 b(will)g(b)r(e)k(the)f(same.)e(Hence)i(all)e(w)n(e)h (need)h(to)523 4281 y(pro)n(v)n(e)26 b(is)845 4451 y SDict begin H.S end 845 4451 a 845 4451 a SDict begin 12 H.A end 845 4451 a 845 4451 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.17) cvn /DEST pdfmark end 845 4451 a FD(E)902 4466 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1198 4359 y Fu(h)1237 4451 y Fv(v)1280 4359 y Fu(\020)1330 4451 y Fv(C)1395 4408 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1389 4476 y Ft(k)q(;\015)1597 4451 y FD(\()p Fv(S;)14 b(\022)r FD(\))1790 4359 y Fu(\021)q(i)1903 4451 y Fs(\025)23 b FD(E)2048 4466 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))2344 4359 y Fu(h)2384 4451 y Fv(v)2427 4359 y Fu(\020)2476 4451 y Fv(C)2541 4408 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2535 4476 y Ft(k)q(;\015)2778 4451 y FD(\()p Fv(S;)14 b(\022)r FD(\))2971 4359 y Fu(\021i)3060 4451 y Fv(:)174 b FD(\(17\))648 4625 y(The)24 b(pro)r(of)f(splits)f(the)i(probabilit)n(y)c(space)k Fv(\022)h FD(:)e(S)g Fs(7!)g FD([0)p Fv(;)14 b FD(1\))24 b(in)n(to)e(subspaces.)h(W)-7 b(e)25 b(pro)n(v)n(e)523 4725 y(that)h(Inequalit)n(y)c(\()1122 4725 y SDict begin H.S end 1122 4725 a Black FD(17)p Black 1206 4672 a SDict begin H.R end 1206 4672 a 1206 4725 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.17) cvn H.B /ANN pdfmark end 1206 4725 a FD(\))k(holds)d(for)i(eac)n(h)g(subspace)f(and)i (therefore)e(it)h(holds)f(for)h(the)g(en)n(tire)523 4825 y(probabilit)n(y)f(space.)648 4924 y(W)-7 b(e)28 b(break)e(the)i (probabilit)n(y)c(space)i(b)n(y)i(conditioning)c(on)j(the)h(follo)n (wing:)p Black 1922 5173 a(17)p Black eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 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y Ft(q)1105 2421 y FD(,)34 b(of)g(the)g(v)-5 b(alues)33 b(of)g(the)i(items)d(mapp)r(ed)h (to)h Fv(\000)12 b FD(.)34 b(In)g(the)g(follo)n(wing)c(w)n(e)523 2549 y(denote)e(this)e(conditional)e(exp)r(ectation)j(b)n(y)g(E)2001 2457 y Fu(h)2040 2549 y Fv(C)2105 2506 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2099 2574 y Ft(k)q(;\015)2307 2549 y FD(\()p Fv(S)5 b FD(\))2427 2457 y Fu(i)2466 2549 y FD(.)28 b(Hence)g(w)n(e)f(get)h(that)995 2815 y SDict begin H.S end 995 2815 a 995 2815 a SDict begin 12 H.A end 995 2815 a 995 2815 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.18) cvn /DEST pdfmark end 995 2815 a FD(E)1052 2723 y Fu(h)1091 2815 y Fv(C)1156 2772 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1150 2840 y Ft(k)q(;\015)1358 2815 y FD(\()p Fv(S)5 b FD(\))1478 2723 y Fu(i)1541 2815 y FD(=)1638 2759 y Fs(j)p Fv(\000)12 b Fs(j)p 1638 2796 109 4 v 1673 2872 a Fv(q)1757 2815 y(v)s FD(\()p Fv(S)1888 2772 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1883 2828 y Fo(>)p Fb(T)2090 2815 y FD(\))1541 3040 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FC([0)p Ft(;)p FC(1\))966 3423 y Fu(h)1005 3516 y Fv(v)1048 3423 y Fu(\020)1098 3516 y Fv(C)1163 3472 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1157 3541 y Ft(k)q(;\015)1399 3516 y FD(\()p Fv(S;)14 b(\022)r FD(\))1592 3423 y Fu(\021i)1704 3516 y FD(in)21 b(a)h(subspace)f (de\014ned)i(b)n(y)e(the)i(conditioning)18 b(ab)r(o)n(v)n(e)523 3673 y(b)n(y)27 b(E)695 3581 y Fu(h)734 3673 y Fv(C)799 3630 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))793 3698 y Ft(k)q(;\015)1035 3673 y FD(\()p Fv(S)5 b FD(\))1155 3581 y Fu(i)1195 3673 y FD(.)27 b(As)h(b)r(efore)f(w)n(e)h(ha)n(v)n(e) 919 3964 y SDict begin H.S end 919 3964 a 919 3964 a SDict begin 12 H.A end 919 3964 a 919 3964 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.19) cvn /DEST pdfmark end 919 3964 a FD(E)976 3872 y Fu(h)1015 3964 y Fv(C)1080 3921 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1074 3989 y Ft(k)q(;\015)1316 3964 y FD(\()p Fv(S)5 b FD(\))1436 3872 y Fu(i)1499 3964 y FD(=)23 b(E)1644 3847 y Fu(\024)1697 3908 y Fs(j)p Fv(\000)1783 3878 y Fq(\003)1821 3908 y Fs(j)p 1697 3945 147 4 v 1750 4021 a Fv(q)1854 3964 y(v)s FD(\()p Fv(S)1985 3921 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1980 3977 y Fo(>)p Fb(T)2187 3964 y FD(\))2219 3847 y Fu(\025)2286 3964 y FD(=)2384 3908 y Fv(v)s FD(\()p Fv(S)2515 3865 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2510 3921 y Fo(>)p Fb(T)2717 3908 y FD(\))p 2384 3945 366 4 v 2546 4021 a Fv(q)2759 3964 y FD(E[)p Fs(j)p Fv(\000)2925 3930 y Fq(\003)2962 3964 y Fs(j)p FD(])249 b(\(19\))523 4224 y(where)35 b(the)i(exp)r(ectation)e(of)g Fv(\000)1544 4193 y Fq(\003)1618 4224 y FD(is)g(tak)n(en)g(o)n(v)n(er)g(all)e(1-1)i (mappings)e(of)j Fv(S)2942 4180 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2937 4236 y Fo(>)p Fb(T)3180 4224 y FD(to)g(the)523 4323 y(arriv)-5 b(al)24 b(times)i Fv(x)1053 4335 y FC(1)1091 4323 y Fv(;)14 b(:)g(:)g(:)f(;)h(x)1322 4335 y Ft(q)1387 4323 y FD(and)27 b(o)n(v)n(er)f Fv(\022)1765 4335 y Ft(D)1825 4323 y FD(.)648 4424 y(W)-7 b(e)29 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Ft(\015)1888 4739 y FD(\()p Fv(X)1989 4751 y Ft(\016)2026 4739 y FD(\))28 b(hence)1445 4924 y SDict begin H.S end 1445 4924 a 1445 4924 a SDict begin 12 H.A end 1445 4924 a 1445 4924 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.20) cvn /DEST pdfmark end 1445 4924 a Fs(j)p Fv(\000)1531 4890 y Fq(\003)1569 4924 y Fs(j)23 b(\024)f FD(min)13 b Fs(f)p Fv(k)20 b Fs(\000)e Fv(\025;)42 b(m)2229 4936 y Ft(\015)2272 4924 y FD(\()p Fv(X)2373 4936 y Ft(\016)2409 4924 y FD(\))p Fs(g)774 b FD(\(20\))p Black 1922 5173 a(18)p Black eop end end %%Page: 19 19 TeXDict begin HPSdict begin 19 18 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.19) cvn /DEST 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418 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3054 486 y Ft(k)q(;\015)3262 461 y FD(\()p Fv(S)5 b FD(\))3382 369 y Fu(i)3445 461 y Fs(\025)523 619 y FD(E)580 526 y Fu(h)619 619 y Fv(C)684 575 y Fq(\003)p FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))678 644 y Ft(k)q(;\015)920 619 y FD(\()p Fv(S)g FD(\))1040 526 y Fu(i)1079 619 y FD(.)27 b(Since)e(this)g(holds)g(for)g(an)n(y)g(subspace)h(de\014ne)g (b)n(y)g(our)f(conditioning)d(on)523 754 y(the)28 b(v)-5 b(alues)26 b(of)i Fv(S)1064 724 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1265 754 y FD(,)g Fv(\016)s FD(,)g Fv(\025)g FD(and)f Fs(f)p Fv(x)1733 766 y FC(1)1771 754 y Fv(;)14 b(:)g(:)g(:)f(;)h(x)2002 766 y Ft(q)2039 754 y Fs(g)27 b FD(the)h(lemma)d(follo)n(ws.)3350 854 y Fs(u)-55 b(t)523 911 y SDict begin H.S end 523 911 a 523 911 a SDict begin 12 H.A end 523 911 a 523 911 a SDict begin [/View [/XYZ H.V]/Dest (lemma.2) cvn /DEST pdfmark end 523 911 a Black 100 x Fr(Lemma)30 b(2.)p Black 40 w Fp(L)l(et)d Fv(S)32 b Fp(b)l(e)c(a)f(set)g(of)i(size)e Fv(n)h Fp(and)f(let)h Fv(Y)46 b Fp(b)l(e)27 b(a)h(subset)f(of)h Fv(S)k Fp(chosen)c(uniformly) 523 1111 y(at)i(r)l(andom)g(amongst)g(al)t(l)h(subsets)e(of)h Fv(S)5 b Fp(.)30 b(Then)h(for)f(any)g FD(25)22 b Fv(<)h(k)j Fs(\024)d Fv(n)p Fp(:)1321 1339 y FD(E)1378 1247 y Fu(h)1416 1339 y Fv(T)1465 1373 y Fs(b)1512 1344 y Fo(k)p 1512 1353 33 3 v 1514 1386 a Fb(2)1554 1373 y Fs(c)1595 1339 y FD(\()p Fv(Y)c FD(\))1726 1247 y Fu(i)1788 1339 y Fs(\025)1876 1222 y Fu(\022)1937 1339 y FD(1)f Fs(\000)2148 1283 y FD(1)p 2090 1320 157 4 v 2090 1407 a(2)2132 1336 y Fs(p)p 2201 1336 46 4 v 71 x Fv(k)2257 1222 y Fu(\023)2336 1339 y Fs(\001)2388 1283 y Fv(T)2437 1295 y Ft(k)2477 1283 y FD(\()p Fv(S)5 b FD(\))p 2388 1320 210 4 v 2472 1396 a(2)p Black 523 1541 a Fp(Pr)l(o)l(of.)p Black 43 w FD(F)-7 b(rom)26 b(Lemma)1298 1541 y SDict begin H.S end 1298 1541 a Black FD(3)p Black 1339 1488 a SDict begin H.R end 1339 1488 a 1339 1541 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.3) cvn H.B /ANN pdfmark end 1339 1541 a 28 w FD(w)n(e)h(kno)n(w)g(that)1068 1783 y(E)1125 1691 y Fu(h)1164 1783 y Fv(T)1213 1817 y Fs(b)1259 1788 y Fo(k)p 1259 1797 33 3 v 1261 1831 a Fb(2)1302 1817 y Fs(c)1343 1783 y FD(\()p Fv(Y)19 b FD(\))1474 1691 y Fu(i)1536 1783 y Fs(\025)k Fv(T)1673 1795 y Ft(k)1713 1783 y FD(\()p Fv(S)5 b FD(\))1889 1680 y Ft(k)1847 1705 y Fu(X)1849 1880 y Ft(r)r FC(=1)1981 1666 y Fu(\022)2042 1727 y Fv(k)2045 1840 y(r)2088 1666 y Fu(\023)2179 1727 y FD(1)p 2159 1764 83 4 v 2159 1840 a(2)2201 1816 y Ft(k)2270 1783 y Fs(\001)2321 1727 y FD(min)o(\()p Fv(r)n(;)14 b Fs(b)p Fv(k)s(=)p FD(2)o Fs(c)p FD(\))p 2321 1764 478 4 v 2537 1840 a Fv(k)2837 1783 y(:)523 2020 y FD(W)-7 b(e)28 b(consider)e(ev)n(en)h(and)g(o)r(dd)h Fv(k)s FD(.)648 2120 y(F)-7 b(or)27 b Fv(k)j FD(ev)n(en:)587 2346 y SDict begin H.S end 587 2346 a 587 2346 a SDict begin 12 H.A end 587 2346 a 587 2346 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.21) cvn /DEST pdfmark end 587 2346 a Fv(T)636 2358 y Ft(k)676 2346 y FD(\()p Fv(S)5 b FD(\))852 2242 y Ft(k)810 2267 y Fu(X)812 2443 y Ft(r)r FC(=1)944 2229 y Fu(\022)1005 2290 y Fv(k)1008 2403 y(r)1051 2229 y Fu(\023)1143 2290 y FD(1)p 1122 2327 83 4 v 1122 2403 a(2)1164 2379 y Ft(k)1233 2346 y Fs(\001)1284 2285 y FD(min)1423 2217 y Fu(\000)1461 2285 y Fv(r)n(;)1533 2217 y Fu(\004)1582 2252 y Ft(k)p 1582 2266 37 4 v 1584 2313 a FC(2)1629 2217 y Fu(\005\001)p 1284 2327 422 4 v 1472 2403 a Fv(k)610 2655 y FD(=)23 b Fv(T)747 2667 y Ft(k)787 2655 y FD(\()p Fv(S)5 b FD(\))907 2488 y Fu(0)907 2637 y(@)988 2544 y Ft(k)q(=)p FC(2)980 2576 y Fu(X)982 2752 y Ft(r)r FC(=1)1114 2538 y Fu(\022)1175 2599 y Fv(k)1178 2712 y(r)1221 2538 y Fu(\023)1295 2599 y Fv(r)p 1292 2636 46 4 v 1292 2712 a(k)1366 2655 y Fs(\001)1438 2599 y FD(1)p 1418 2636 83 4 v 1418 2712 a(2)1460 2688 y Ft(k)1529 2655 y FD(+)1729 2551 y Ft(k)1687 2576 y Fu(X)1612 2758 y Ft(r)r FC(=)p Ft(k)q(=)p FC(2+1)1897 2538 y Fu(\022)1958 2599 y Fv(k)1961 2712 y(r)2004 2538 y Fu(\023)2095 2599 y FD(1)p 2075 2636 V 2075 2712 a(2)2117 2688 y Ft(k)2186 2655 y Fs(\001)2237 2599 y FD(1)p 2237 2636 42 4 v 2237 2712 a(2)2289 2488 y Fu(1)2289 2637 y(A)610 2987 y FD(=)23 b Fv(T)747 2999 y Ft(k)787 2987 y FD(\()p Fv(S)5 b FD(\))907 2820 y Fu(0)907 2969 y(@)990 2931 y FD(1)p 990 2968 V 990 3044 a(2)1064 2876 y Ft(k)q(=)p FC(2)1056 2908 y Fu(X)1057 3084 y Ft(r)r FC(=1)1189 2870 y Fu(\022)1250 2931 y Fv(k)22 b Fs(\000)c FD(1)1253 3044 y Fv(r)k Fs(\000)c FD(1)1439 2870 y Fu(\023)1573 2931 y FD(1)p 1510 2968 168 4 v 1510 3044 a(2)1552 3020 y Ft(k)q Fq(\000)p FC(1)1706 2987 y FD(+)1799 2931 y(1)p 1799 2968 42 4 v 1799 3044 a(2)1869 2987 y Fs(\001)h FD(Prob)2087 2870 y Fu(\024)2145 2987 y Fv(B)t(in)2291 2870 y Fu(\022)2352 2987 y Fv(k)s(;)2444 2931 y FD(1)p 2444 2968 V 2444 3044 a(2)2496 2870 y Fu(\023)2580 2987 y Fs(\025)2677 2931 y Fv(k)p 2677 2968 46 4 v 2679 3044 a FD(2)2752 2987 y(+)f(1)2891 2870 y Fu(\025)2934 2820 y(1)2934 2969 y(A)610 3319 y FD(=)708 3263 y Fv(T)757 3275 y Ft(k)797 3263 y FD(\()p Fv(S)5 b FD(\))p 708 3300 210 4 v 792 3376 a(2)928 3152 y Fu(0)928 3302 y(@)1000 3208 y Ft(k)q(=)p FC(2)p Fq(\000)p FC(1)1035 3240 y Fu(X)1036 3416 y Ft(r)r FC(=0)1203 3202 y Fu(\022)1264 3263 y Fv(k)21 b Fs(\000)d FD(1)1338 3376 y Fv(r)1453 3202 y Fu(\023)1587 3263 y FD(1)p 1524 3300 168 4 v 1524 3376 a(2)1566 3352 y Ft(k)q Fq(\000)p FC(1)1720 3319 y FD(+)g(Prob)1979 3202 y Fu(\024)2037 3319 y Fv(B)t(in)2183 3202 y Fu(\022)2244 3319 y Fv(k)s(;)2336 3263 y FD(1)p 2336 3300 42 4 v 2336 3376 a(2)2388 3202 y Fu(\023)2472 3319 y Fs(\025)2570 3263 y Fv(k)p 2570 3300 46 4 v 2572 3376 a FD(2)2644 3319 y(+)g(1)2783 3202 y Fu(\025)2826 3152 y(1)2826 3302 y(A)610 3601 y FD(=)708 3545 y Fv(T)757 3557 y Ft(k)797 3545 y FD(\()p Fv(S)5 b FD(\))p 708 3582 210 4 v 792 3658 a(2)928 3484 y Fu(\022)989 3601 y FD(Prob)1165 3484 y Fu(\024)1223 3601 y Fv(B)t(in)1369 3484 y Fu(\022)1430 3601 y Fv(k)21 b Fs(\000)d FD(1)p Fv(;)1665 3545 y FD(1)p 1665 3582 42 4 v 1665 3658 a(2)1717 3484 y Fu(\023)1801 3601 y Fs(\024)1899 3545 y Fv(k)p 1899 3582 46 4 v 1901 3658 a FD(2)1973 3601 y Fs(\000)g FD(1)2112 3484 y Fu(\025)2173 3601 y FD(+)h(Prob)2433 3484 y Fu(\024)2491 3601 y Fv(B)t(in)2637 3484 y Fu(\022)2698 3601 y Fv(k)s(;)2790 3545 y FD(1)p 2790 3582 42 4 v 2790 3658 a(2)2842 3484 y Fu(\023)2926 3601 y Fs(\025)3023 3545 y Fv(k)p 3023 3582 46 4 v 3025 3658 a FD(2)3098 3601 y(+)f(1)3237 3484 y Fu(\025)o(\023)610 3834 y FD(=)708 3778 y Fv(T)757 3790 y Ft(k)797 3778 y FD(\()p Fv(S)5 b FD(\))p 708 3815 210 4 v 792 3891 a(2)928 3717 y Fu(\022)999 3778 y FD(1)p 999 3815 42 4 v 999 3891 a(2)1069 3834 y(+)1162 3778 y(1)p 1162 3815 V 1162 3891 a(2)1231 3834 y Fs(\000)1315 3717 y Fu(\022)1417 3778 y Fv(k)1376 3891 y(k)s(=)p FD(2)1504 3717 y Fu(\023)1638 3778 y FD(1)p 1576 3815 167 4 v 1576 3891 a(2)1618 3867 y Ft(k)q FC(+1)1752 3717 y Fu(\023)p Black 3257 3834 a FD(\(21\))p Black 610 4066 a(=)708 4010 y Fv(T)757 4022 y Ft(k)797 4010 y FD(\()p Fv(S)g FD(\))p 708 4047 210 4 v 792 4123 a(2)928 3949 y Fu(\022)989 4066 y FD(1)18 b Fs(\000)1132 3949 y Fu(\022)1234 4010 y Fv(k)1193 4123 y(k)s(=)p FD(2)1322 3949 y Fu(\023)1455 4010 y FD(1)p 1393 4047 167 4 v 1393 4123 a(2)1435 4099 y Ft(k)q FC(+1)1569 3949 y Fu(\023)1630 4066 y Fv(:)p Black 1604 w FD(\(22\))p Black 523 4281 a(W)-7 b(e)28 b(deriv)n(e)e(\()943 4281 y SDict begin H.S end 943 4281 a Black FD(21)p Black 1026 4228 a SDict begin H.R end 1026 4228 a 1026 4281 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.21) cvn H.B /ANN pdfmark end 1026 4281 a FD(\))i(as)f(follo)n(ws:)p Black 575 4430 a Fr({)p Black 41 w FD(Prob)841 4362 y Fu(\002)889 4430 y Fv(B)t(in)1035 4362 y Fu(\000)1073 4430 y Fv(k)21 b Fs(\000)d FD(1)p Fv(;)1309 4397 y FC(1)p 1309 4411 34 4 v 1309 4458 a(2)1352 4362 y Fu(\001)1413 4430 y Fs(\024)1510 4397 y Ft(k)p 1510 4411 37 4 v 1512 4458 a FC(2)1575 4430 y Fs(\000)g FD(1)1714 4362 y Fu(\003)1771 4430 y FD(=)23 b(1)p Fv(=)p FD(2)p Fv(:)664 4526 y FD(Consider)43 b Fv(k)33 b Fs(\000)c FD(1)44 b(coin)g(tosses)f(of)i(a)f(fair)g(coin,)f (the)i(n)n(um)n(b)r(er)e(of)i(heads)f(can)h(b)r(e)664 4625 y(0)p Fv(;)14 b FD(1)p Fv(;)g(:)g(:)g(:)f(;)h(k)23 b Fs(\000)d FD(1.)31 b(The)f(probabilit)n(y)d(that)k(there)g(are)f Fv(i)g FD(heads)g(is)g(equal)f(to)i(the)g(prob-)664 4725 y(abilit)n(y)k(that)k(there)f(are)g Fv(k)28 b Fs(\000)e FD(1)f Fs(\000)g Fv(i)38 b FD(heads.)g(As)h Fv(k)28 b Fs(\000)e FD(1)f Fs(\000)g FD(\()p Fv(k)s(=)p FD(2)g Fs(\000)g FD(1\))41 b(=)g Fv(k)s(=)p FD(2,)d(the)664 4825 y(set)c Fs(f)p FD(0)p Fv(;)14 b(:)g(:)g(:)e(k)s(=)p FD(2)22 b Fs(\000)g FD(1)p Fs(g)33 b FD(is)f(disjoin)n(t)g(from)g(the)j (set)e Fs(f)p Fv(k)s(=)p FD(2)p Fv(;)14 b(:)g(:)g(:)e(;)i(k)s Fs(g)33 b FD(and)h(their)e(union)h(is)664 4924 y Fs(f)p FD(0)p Fv(;)14 b(:)g(:)g(:)f(;)h(k)21 b Fs(\000)d FD(1)p Fs(g)p FD(.)p Black 1922 5173 a(19)p Black eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end 515 125 a Black Black Black 575 448 a Fr({)p Black 41 w FD(Prob)841 381 y Fu(\002)889 448 y Fv(B)t(in)1035 381 y Fu(\000)1073 448 y Fv(k)s(;)1166 416 y FC(1)p 1166 430 34 4 v 1166 477 a(2)1209 381 y Fu(\001)1270 448 y Fs(\025)1367 416 y Ft(k)p 1367 430 37 4 v 1369 477 a FC(2)1432 448 y FD(+)18 b(1)1571 381 y Fu(\003)1628 448 y FD(=)1726 416 y FC(1)p 1726 430 34 4 v 1726 477 a(2)1788 448 y Fs(\000)1871 381 y Fu(\000)1942 411 y Ft(k)1909 477 y(k)q(=)p FC(2)2012 381 y Fu(\001)2114 416 y FC(1)p 2060 430 141 4 v 2060 478 a(2)2093 461 y Fo(k)q Fb(+1)2211 448 y Fv(:)664 558 y FD(Consider)28 b Fv(k)k FD(tosses)c(of)i(a)f(fair)f(coin,)g(the)i (outcome)e(can)h(ha)n(v)n(e)f(0)p Fv(;)14 b FD(1)p Fv(;)g(:)g(:)g(:)f (;)h(k)32 b FD(heads)d(\(an)664 657 y(o)r(dd)h(n)n(um)n(b)r(er)e(of)i (outcomes\).)e(As)h(ab)r(o)n(v)n(e,)g(the)h(probabilit)n(y)25 b(that)30 b(there)f(b)r(e)h Fv(i)f FD(heads,)664 757 y Fv(i)23 b FD(=)f(0)p Fv(;)14 b FD(1)p Fv(;)g(:)g(:)g(:)f(;)h(k)s(=)p FD(2)5 b Fs(\000)g FD(1)18 b(is)h(equal)g(to)i(the)g(probabilit)n(y)16 b(that)21 b(there)g(b)r(e)g Fv(k)7 b Fs(\000)e Fv(i)20 b FD(heads.)g(Ergo,)664 857 y(Prob)o([)14 b(#)28 b(heads)f Fs(2)c FD(0)p Fv(;)14 b(:)g(:)g(:)f(;)h(k)s(=)p FD(2)j Fs(\000)h FD(1)c(])23 b(=)g(Prob)n([)14 b(#)28 b(heads)f Fs(2)c Fv(k)s(=)p FD(2)18 b(+)g(1)p Fv(;)c(:)g(:)g(:)f(;)h(k)i FD(],)28 b(and)1400 1018 y(Prob)n([)14 b(#)28 b(heads)f Fs(2)c FD(0)p Fv(;)14 b(:)g(:)g(:)g(;)g(k)s(=)p FD(2)j Fs(\000)h FD(1)13 b(])1446 1142 y(+Prob)n([)h(#)28 b(heads)f Fs(2)d Fv(k)s(=)p FD(2)17 b(+)h(1)p Fv(;)c(:)g(:)g(:)f(;)h(k)j FD(])1446 1267 y(+Prob)n([)d(#)28 b(heads)f(=)c Fv(k)s(=)p FD(2)12 b(])24 b(=)e(1)p Fv(:)664 1428 y FD(As)33 b(the)g(probabilit)n (y)c(that)34 b(there)e(are)g(exactly)f Fv(k)s(=)p FD(2)h(heads)h(is) 2695 1361 y Fu(\000)2767 1391 y Ft(k)2733 1457 y(k)q(=)p FC(2)2837 1361 y Fu(\001)2903 1395 y FC(1)p 2885 1409 70 4 v 2885 1458 a(2)2918 1441 y Fo(k)2965 1428 y FD(,)g(solving)d(for) 664 1542 y(Prob)o([)14 b(#)28 b(heads)f Fs(2)c Fv(k)s(=)p FD(2)17 b(+)h(1)p Fv(;)c(:)g(:)g(:)g(;)g(k)i FD(])28 b(giv)n(es)d(the)j(desired)e(result.)648 1691 y(F)-7 b(rom)26 b(Stirling's)e(form)n(ula)h(w)n(e)i(get:)947 1915 y SDict begin H.S end 947 1915 a 947 1915 a SDict begin 12 H.A end 947 1915 a 947 1915 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.23) cvn /DEST pdfmark end 947 1915 a -117 x Fu(\022)1049 1859 y Fv(k)1008 1972 y(k)s(=)p FD(2)1137 1798 y Fu(\023)1228 1859 y FD(1)p 1208 1896 83 4 v 1208 1972 a(2)1250 1948 y Ft(k)1323 1915 y Fs(\024)1665 1854 y Fv(e)1704 1783 y Fs(p)p 1773 1783 46 4 v 71 x Fv(k)s FD(\()1861 1822 y Ft(k)p 1861 1836 37 4 v 1864 1883 a(e)1908 1854 y FD(\))1940 1824 y Ft(k)p 1421 1896 806 4 v 1421 1926 a Fu(\022)1482 1975 y Fs(p)p 1551 1975 92 4 v 68 x FD(2)p Fv(\031)1643 1946 y Fu(q)p 1726 1946 57 4 v 1736 2011 a Ft(k)p 1736 2025 37 4 v 1738 2072 a FC(2)1782 1976 y Fu(\000\000)1869 2011 y Ft(k)p 1869 2025 V 1871 2072 a FC(2)1915 1976 y Fu(\001)1953 2043 y Fv(=e)2034 1976 y Fu(\001)2081 1969 y Fo(k)p 2081 1978 33 3 v 2083 2011 a Fb(2)2128 1926 y Fu(\023)2189 1944 y FC(2)2254 1915 y Fs(\001)2326 1859 y FD(1)p 2306 1896 83 4 v 2306 1972 a(2)2348 1948 y Ft(k)2421 1915 y FD(=)2520 1859 y Fv(e)2559 1788 y Fs(p)p 2628 1788 46 4 v 71 x Fv(k)p 2519 1896 157 4 v 2519 1972 a(\031)22 b Fs(\001)c Fv(k)2708 1915 y(<)2842 1859 y FD(1)p 2806 1896 116 4 v 2806 1913 a Fs(p)p 2875 1913 46 4 v 71 x Fv(k)2958 1915 y(:)276 b FD(\(23\))523 2254 y(By)27 b(substituting)f(\()1147 2254 y SDict begin H.S end 1147 2254 a Black FD(23)p Black 1231 2201 a SDict begin H.R end 1231 2201 a 1231 2254 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.23) cvn H.B /ANN pdfmark end 1231 2254 a FD(\))i(in)f(\()1420 2254 y SDict begin H.S end 1420 2254 a Black FD(22)p Black 1503 2201 a SDict begin H.R end 1503 2201 a 1503 2254 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.21) cvn H.B /ANN pdfmark end 1503 2254 a FD(\))h(w)n(e)f(get)1379 2483 y Fv(T)1428 2498 y Fq(b)p Ft(k)q(=)p FC(2)q Fq(c)1596 2483 y FD(\()p Fv(S)5 b FD(\))24 b Fs(\025)1837 2427 y Fv(T)1886 2439 y Ft(k)1926 2427 y FD(\()p Fv(S)5 b FD(\))p 1837 2464 210 4 v 1921 2540 a(2)2057 2366 y Fu(\022)2118 2483 y FD(1)18 b Fs(\000)2328 2427 y FD(1)p 2271 2464 157 4 v 2271 2551 a(2)2313 2480 y Fs(p)p 2381 2480 46 4 v 2381 2551 a Fv(k)2437 2366 y Fu(\023)2526 2483 y Fv(;)523 2665 y FD(\014nishing)26 b(the)i(pro)r(of)f(for)g Fv(k)j FD(ev)n(en.)648 2764 y(F)-7 b(or)27 b(an)g(o)r(dd)h Fv(k)s FD(:)523 2987 y SDict begin H.S end 523 2987 a 523 2987 a SDict begin 12 H.A end 523 2987 a 523 2987 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.24) cvn /DEST pdfmark end 523 2987 a Fv(T)572 2999 y Ft(k)612 2987 y FD(\()p Fv(S)5 b FD(\))788 2883 y Ft(k)747 2908 y Fu(X)748 3083 y Ft(r)r FC(=1)880 2869 y Fu(\022)941 2930 y Fv(k)944 3043 y(r)987 2869 y Fu(\023)1079 2930 y FD(1)p 1058 2967 83 4 v 1058 3043 a(2)1100 3020 y Ft(k)1161 2925 y FD(min)n(\()p Fv(r)n(;)1403 2858 y Fu(\004)1452 2892 y Ft(k)p 1452 2906 37 4 v 1454 2954 a FC(2)1499 2858 y Fu(\005)1538 2925 y FD(\))p 1161 2967 411 4 v 1343 3043 a Fv(k)546 3320 y FD(=)23 b Fv(T)683 3332 y Ft(k)723 3320 y FD(\()p Fv(S)5 b FD(\))843 3128 y Fu(0)843 3274 y(B)843 3328 y(@)916 3203 y Fs(b)963 3174 y Fo(k)p 963 3183 33 3 v 965 3217 a Fb(2)1005 3203 y Fs(c)919 3241 y Fu(X)921 3417 y Ft(r)r FC(=1)1056 3203 y Fu(\022)1117 3264 y Fv(k)1120 3377 y(r)1163 3203 y Fu(\023)1237 3264 y Fv(r)p 1234 3301 46 4 v 1234 3377 a(k)1308 3320 y Fs(\001)1380 3264 y FD(1)p 1360 3301 83 4 v 1360 3377 a(2)1402 3353 y Ft(k)1471 3320 y FD(+)1554 3203 y Fu(\026)1612 3264 y Fv(k)p 1612 3301 46 4 v 1614 3377 a FD(2)1668 3203 y Fu(\027)1735 3320 y Fs(\001)1788 3264 y FD(1)p 1786 3301 V 1786 3377 a Fv(k)1984 3216 y Ft(k)1943 3241 y Fu(X)1856 3436 y Ft(r)r FC(=)1940 3442 y Fs(b)1986 3413 y Fo(k)p 1986 3422 33 3 v 1988 3456 a Fb(2)2028 3442 y Fs(c)2065 3436 y FC(+1)2163 3203 y Fu(\022)2224 3264 y Fv(k)2227 3377 y(r)2270 3203 y Fu(\023)2362 3264 y FD(1)p 2341 3301 83 4 v 2341 3377 a(2)2383 3353 y Ft(k)2433 3128 y Fu(1)2433 3274 y(C)2433 3328 y(A)546 3702 y Fs(\025)23 b Fv(T)683 3714 y Ft(k)723 3702 y FD(\()p Fv(S)5 b FD(\))843 3510 y Fu(0)843 3656 y(B)843 3709 y(@)926 3646 y FD(1)p 926 3683 42 4 v 926 3759 a(2)992 3585 y Fs(b)1039 3556 y Fo(k)p 1039 3565 33 3 v 1041 3599 a Fb(2)1081 3585 y Fs(c)995 3623 y Fu(X)996 3799 y Ft(r)r FC(=1)1131 3585 y Fu(\022)1192 3646 y Fv(k)22 b Fs(\000)c FD(1)1196 3759 y Fv(r)j Fs(\000)d FD(1)1381 3585 y Fu(\023)1515 3646 y FD(1)p 1452 3683 168 4 v 1452 3759 a(2)1494 3735 y Ft(k)q Fq(\000)p FC(1)1676 3702 y FD(+)1796 3646 y Fv(k)k Fs(\000)c FD(1)p 1796 3683 189 4 v 1847 3759 a(2)p Fv(k)2137 3598 y Ft(k)2096 3623 y Fu(X)2009 3818 y Ft(r)r FC(=)2093 3824 y Fs(b)2140 3795 y Fo(k)p 2140 3804 33 3 v 2142 3838 a Fb(2)2182 3824 y Fs(c)2219 3818 y FC(+1)2316 3585 y Fu(\022)2378 3646 y Fv(k)2381 3759 y(r)2423 3585 y Fu(\023)2515 3646 y FD(1)p 2494 3683 83 4 v 2494 3759 a(2)2536 3735 y Ft(k)2587 3510 y Fu(1)2587 3656 y(C)2587 3709 y(A)546 4009 y FD(=)644 3953 y Fv(T)693 3965 y Ft(k)733 3953 y FD(\()p Fv(S)5 b FD(\))p 644 3990 210 4 v 728 4066 a(2)864 3892 y Fu(\022)925 4009 y FD(Prob)1101 3892 y Fu(\024)1159 4009 y Fv(B)t(in)1305 3892 y Fu(\022)1366 4009 y Fv(k)21 b Fs(\000)d FD(1)p Fv(;)1601 3953 y FD(1)p 1601 3990 42 4 v 1601 4066 a(2)1653 3892 y Fu(\023)1737 4009 y Fs(\024)1825 3892 y Fu(\026)1883 3953 y Fv(k)p 1883 3990 46 4 v 1885 4066 a FD(2)1939 3892 y Fu(\027)2006 4009 y Fs(\000)g FD(1)2145 3892 y Fu(\025)2206 4009 y FD(+)2299 3953 y Fv(k)k Fs(\000)c FD(1)p 2299 3990 189 4 v 2371 4066 a Fv(k)2498 4009 y FD(Prob)2675 3892 y Fu(\024)2733 4009 y Fv(B)t(in)2879 3892 y Fu(\022)2939 4009 y Fv(k)s(;)3032 3953 y FD(1)p 3032 3990 42 4 v 3032 4066 a(2)3083 3892 y Fu(\023)3168 4009 y Fs(\025)3255 3892 y Fu(\026)3314 3953 y Fv(k)p 3314 3990 46 4 v 3316 4066 a FD(2)3369 3892 y Fu(\027)3436 4009 y FD(+)g(1)3575 3892 y Fu(\025)o(\023)546 4242 y FD(=)644 4185 y Fv(T)693 4197 y Ft(k)733 4185 y FD(\()p Fv(S)5 b FD(\))p 644 4223 210 4 v 728 4299 a(2)864 4125 y Fu(\022\022)996 4185 y FD(1)p 996 4223 42 4 v 996 4299 a(2)1066 4242 y Fs(\000)1149 4125 y Fu(\022)1284 4185 y Fv(k)21 b Fs(\000)d FD(1)1210 4299 y(\()p Fv(k)j Fs(\000)d FD(1\))p Fv(=)p FD(2)1546 4125 y Fu(\023)1638 4185 y FD(1)p 1617 4223 83 4 v 1617 4299 a(2)1659 4275 y Ft(k)1710 4125 y Fu(\023)1789 4242 y FD(+)1882 4185 y Fv(k)k Fs(\000)c FD(1)p 1882 4223 189 4 v 1954 4299 a Fv(k)2099 4242 y Fs(\001)2151 4185 y FD(1)p 2151 4223 42 4 v 2151 4299 a(2)2202 4125 y Fu(\023)546 4474 y FD(=)644 4418 y Fv(T)693 4430 y Ft(k)733 4418 y FD(\()p Fv(S)5 b FD(\))p 644 4455 210 4 v 728 4531 a(2)864 4357 y Fu(\022)925 4474 y FD(1)18 b Fs(\000)1068 4357 y Fu(\022\022)1264 4418 y Fv(k)j Fs(\000)d FD(1)1190 4531 y(\()p Fv(k)k Fs(\000)c FD(1\))p Fv(=)p FD(2)1526 4357 y Fu(\023)1618 4418 y FD(1)p 1597 4455 83 4 v 1597 4531 a(2)1639 4507 y Ft(k)1708 4474 y FD(+)1824 4418 y(1)p 1801 4455 88 4 v 1801 4531 a(2)p Fv(k)1899 4357 y Fu(\023\023)2021 4474 y Fv(:)p Black 1488 w FD(\(24\))p Black 523 4681 a(Lemma)819 4681 y SDict begin H.S end 819 4681 a Black FD(4)p Black 861 4627 a SDict begin H.R end 861 4627 a 861 4681 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.4) cvn H.B /ANN pdfmark end 861 4681 a 28 w FD(sho)n(ws)26 b(b)n(y)h(induction)f(that)i(for)f(an)n(y)g(o)r(dd)h Fv(k)e(>)c FD(25)1410 4887 y SDict begin H.S end 1410 4887 a 1410 4887 a SDict begin 12 H.A end 1410 4887 a 1410 4887 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.25) cvn /DEST pdfmark end 1410 4887 a -117 x Fu(\022)1545 4831 y Fv(k)f Fs(\000)d FD(1)1471 4944 y(\()p Fv(k)k Fs(\000)c FD(1\))p Fv(=)p FD(2)1808 4770 y Fu(\023)1899 4831 y FD(1)p 1879 4868 83 4 v 1879 4944 a(2)1921 4920 y Ft(k)1990 4887 y FD(+)2105 4831 y(1)p 2083 4868 88 4 v 2083 4944 a(2)p Fv(k)2203 4887 y Fs(\024)2358 4831 y FD(1)p 2301 4868 157 4 v 2301 4955 a(2)2343 4885 y Fs(p)p 2411 4885 46 4 v 2411 4955 a Fv(k)2495 4887 y(:)739 b FD(\(25\))p Black 1922 5173 a(20)p Black eop end end %%Page: 21 21 TeXDict begin HPSdict begin 21 20 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.21) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(\(Note)28 b(that)g(b)n(y)f(\()1083 448 y SDict begin H.S end 1083 448 a Black FD(23)p Black 1166 395 a SDict begin H.R end 1166 395 a 1166 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.23) cvn H.B /ANN pdfmark end 1166 448 a FD(\))h(w)n(e)g(get)f(that)1174 565 y Fu(\022)1309 626 y Fv(k)21 b Fs(\000)d FD(1)1235 739 y(\()p Fv(k)k Fs(\000)c FD(1\))p Fv(=)p FD(2)1572 565 y Fu(\023)1663 626 y FD(1)p 1643 663 83 4 v 1643 739 a(2)1685 715 y Ft(k)1754 682 y FD(+)1870 626 y(1)p 1847 663 88 4 v 1847 739 a(2)p Fv(k)1967 682 y Fs(\024)2065 626 y FD(1)p 2065 663 42 4 v 2065 739 a(2)2116 565 y Fu(\022)2193 626 y Fv(e)p 2187 663 51 4 v 2187 739 a(\031)2366 626 y FD(1)p 2257 663 259 4 v 2257 680 a Fs(p)p 2327 680 189 4 v 2327 747 a Fv(k)j Fs(\000)d FD(1)2544 682 y(+)2639 626 y(1)p 2637 663 46 4 v 2637 739 a Fv(k)2693 565 y Fu(\023)523 932 y FD(whic)n(h)27 b(is)f(smaller)e(than)j(1)p Fv(=)p FD(\(2)1482 861 y Fs(p)p 1550 861 V 1550 932 a Fv(k)s FD(\))h(for)f Fv(k)f Fs(\025)d FD(63.\))648 1032 y(Substituting)j(\()1155 1032 y SDict begin H.S end 1155 1032 a Black FD(25)p Black 1238 979 a SDict begin H.R end 1238 979 a 1238 1032 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.25) cvn H.B /ANN pdfmark end 1238 1032 a FD(\))i(in)n(to)f (\()1499 1032 y SDict begin H.S end 1499 1032 a Black FD(24)p Black 1582 979 a SDict begin H.R end 1582 979 a 1582 1032 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.24) cvn H.B /ANN pdfmark end 1582 1032 a FD(\))h(w)n(e)f(get)1379 1266 y Fv(T)1428 1281 y Fq(b)p Ft(k)q(=)p FC(2)q Fq(c)1596 1266 y FD(\()p Fv(S)5 b FD(\))24 b Fs(\025)1837 1210 y Fv(T)1886 1222 y Ft(k)1926 1210 y FD(\()p Fv(S)5 b FD(\))p 1837 1247 210 4 v 1921 1323 a(2)2057 1149 y Fu(\022)2118 1266 y FD(1)18 b Fs(\000)2328 1210 y FD(1)p 2271 1247 157 4 v 2271 1335 a(2)2313 1264 y Fs(p)p 2381 1264 46 4 v 2381 1335 a Fv(k)2437 1149 y Fu(\023)2526 1266 y Fv(;)523 1496 y FD(\014nishing)26 b(the)i(pro)r(of)f(for)g Fv(k)j FD(o)r(dd.)3350 1596 y Fs(u)-55 b(t)523 1674 y SDict begin H.S end 523 1674 a 523 1674 a SDict begin 12 H.A end 523 1674 a 523 1674 a SDict begin [/View [/XYZ H.V]/Dest (lemma.3) cvn /DEST pdfmark end 523 1674 a Black 100 x Fr(Lemma)30 b(3.)p Black 40 w Fp(L)l(et)k Fv(S)i FD(=)30 b Fs(f)p Fv(s)1388 1786 y FC(1)1456 1774 y Fv(>)h(s)1591 1786 y FC(2)1659 1774 y Fv(>)g Fs(\001)14 b(\001)g(\001)31 b Fv(>)f(s)2017 1786 y Ft(n)2062 1774 y Fs(g)k Fp(and)h(let)f Fv(S)2481 1744 y Fq(0)2538 1774 y Fp(b)l(e)g(a)h(subset)e(of)i Fv(S)k Fp(chosen)523 1874 y(uniformly)27 b(at)f(r)l(andom)h(amongst)g(al)t(l)g(subsets)e(of)i Fv(S)5 b Fp(.)27 b(L)l(et)e Fv(t;)14 b(k)29 b Fp(b)l(e)e(inte)l(gers)f (s.t.)g Fv(t)d Fs(\024)g Fv(k)j Fs(\024)d Fv(n)p Fp(.)523 1973 y(Then)1221 2236 y FD(E[[)p Fv(T)1373 2248 y Ft(t)1401 2236 y FD(\()p Fv(S)1489 2202 y Fq(0)1513 2236 y FD(\)]])g Fs(\025)g Fv(T)1751 2248 y Ft(k)1791 2236 y FD(\()p Fv(S)5 b FD(\))1967 2133 y Ft(k)1925 2157 y Fu(X)1927 2333 y Ft(r)r FC(=1)2059 2119 y Fu(\022)2120 2180 y Fv(k)2123 2293 y(r)2166 2119 y Fu(\023)2257 2180 y FD(1)p 2237 2217 83 4 v 2237 2293 a(2)2279 2269 y Ft(k)2339 2180 y FD(min)o(\()p Fv(r)n(;)14 b(t)p FD(\))p 2339 2217 305 4 v 2469 2293 a Fv(k)2684 2236 y(:)p Black 523 2490 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Let)29 b Fv(R)c FD(=)f Fs(f)p Fv(s)1191 2502 y FC(1)1253 2490 y Fv(>)g(:)14 b(:)g(:)24 b(>)h(s)1592 2502 y Ft(k)1632 2490 y Fs(g)j FD(b)r(e)h(the)g Fv(k)j FD(elemen)n(ts)26 b(of)j(largest)d(v)-5 b(alue)27 b(in)h Fv(S)5 b FD(.)28 b(Condi-)523 2590 y(tioned)h(up)r(on) h Fs(j)p Fv(S)1070 2560 y Fq(0)1111 2590 y Fs(\\)19 b Fv(R)q Fs(j)26 b FD(=)h Fv(r)r FD(,)j(the)g(exp)r(ectation)f(of)g Fv(v)s FD(\()p Fv(S)2304 2560 y Fq(0)2348 2590 y Fs(\\)20 b Fv(R)q FD(\))30 b(is)e Fv(r)r(=k)33 b FD(times)28 b(the)i(sum)f(of) 523 2690 y(the)f(v)-5 b(alues)26 b(in)h Fv(R)h FD(whic)n(h)f(is)f Fv(T)1471 2702 y Ft(k)1512 2690 y FD(\()p Fv(S)5 b FD(\).)648 2790 y(The)32 b(lemma)c(no)n(w)k(follo)n(ws)c(b)n(y)k(summing)d(o)n(v)n (er)h(all)g(p)r(ossible)f(v)-5 b(alues)31 b(of)g Fv(r)k FD(using)c(the)523 2890 y(follo)n(wing)24 b(t)n(w)n(o)i(facts:)523 2975 y SDict begin H.S end 523 2975 a 523 2975 a SDict begin 12 H.A end 523 2975 a 523 2975 a SDict begin [/View [/XYZ H.V]/Dest (Item.9) cvn /DEST pdfmark end 523 2975 a Black 558 3059 a FD(1.)p Black 41 w(The)i(probabilit)n(y)23 b(that)28 b Fs(j)p Fv(S)1518 3029 y Fq(0)1560 3059 y Fs(\\)19 b Fv(R)p Fs(j)k FD(=)g Fv(r)30 b FD(is)1982 2992 y Fu(\000)2020 3022 y Ft(k)2022 3088 y(r)2057 2992 y Fu(\001)2123 3026 y FC(1)p 2105 3040 70 4 v 2105 3089 a(2)2138 3072 y Fo(k)2184 3059 y FD(.)523 3089 y SDict begin H.S end 523 3089 a 523 3089 a SDict begin 12 H.A end 523 3089 a 523 3089 a SDict begin [/View [/XYZ H.V]/Dest (Item.10) cvn /DEST pdfmark end 523 3089 a Black 558 3160 a FD(2.)p Black 41 w(If)h Fv(t)c Fs(\024)h Fv(r)33 b FD(then)e Fv(T)1211 3172 y Ft(t)1239 3160 y FD(\()p Fv(S)1327 3129 y Fq(0)1351 3160 y FD(\))c Fs(\025)h Fv(T)1552 3172 y Ft(t)1580 3160 y FD(\()p Fv(S)1668 3129 y Fq(0)1712 3160 y Fs(\\)20 b Fv(R)q FD(\))28 b Fs(\025)2017 3127 y Ft(t)p 2013 3141 33 4 v 2013 3188 a(r)2056 3160 y Fv(v)s FD(\()p Fv(S)2187 3129 y Fq(0)2230 3160 y Fs(\\)21 b Fv(R)q FD(\))30 b(and)g(if)g Fv(t)e Fs(\025)f Fv(r)33 b FD(than)d Fv(T)3140 3172 y Ft(t)3169 3160 y FD(\()p Fv(S)3257 3129 y Fq(0)3281 3160 y FD(\))d Fs(\025)664 3259 y Fv(v)s FD(\()p Fv(S)795 3229 y Fq(0)837 3259 y Fs(\\)19 b Fv(R)q FD(\).)3350 3360 y Fs(u)-55 b(t)523 3437 y SDict begin H.S end 523 3437 a 523 3437 a SDict begin 12 H.A end 523 3437 a 523 3437 a SDict begin [/View [/XYZ H.V]/Dest (lemma.4) cvn /DEST pdfmark end 523 3437 a Black 100 x Fr(Lemma)30 b(4.)p Black 40 w Fp(F)-6 b(or)30 b(any)g(o)l(dd)h Fv(k)26 b(>)d FD(25)1436 3650 y Fu(\022)1570 3711 y Fv(k)f Fs(\000)c FD(1)1497 3824 y(\()p Fv(k)j Fs(\000)d FD(1\))p Fv(=)p FD(2)1833 3650 y Fu(\023)1925 3711 y FD(1)p 1904 3748 83 4 v 1904 3824 a(2)1946 3800 y Ft(k)2015 3767 y FD(+)2131 3711 y(1)p 2108 3748 88 4 v 2108 3824 a(2)p Fv(k)2228 3767 y Fs(\024)2383 3711 y FD(1)p 2326 3748 157 4 v 2326 3835 a(2)2368 3764 y Fs(p)p 2437 3764 46 4 v 71 x Fv(k)p Black 523 3997 a Fp(Pr)l(o)l(of.)p Black 43 w FD(W)-7 b(e)28 b(pro)n(v)n(e)e(the)i (lemma)d(b)n(y)i(induction)f(on)h Fv(k)648 4098 y FD(Note)g(that)h (this)f(inequalit)n(y)d(do)r(esn't)k(hold)e(for)h Fv(k)f Fs(\024)d FD(25.)648 4198 y(F)-7 b(or)27 b(simplicit)n(y)-7 b(,)22 b(w)n(e)28 b(pro)n(v)n(e)e(the)i(equiv)-5 b(alen)n(t)25 b(inequalit)n(y:)1334 4311 y Fu(\022\022)1530 4372 y Fv(k)c Fs(\000)d FD(1)1456 4485 y(\()p Fv(k)j Fs(\000)e FD(1\))p Fv(=)p FD(2)1792 4311 y Fu(\023)1884 4372 y FD(1)p 1863 4409 83 4 v 1863 4485 a(2)1905 4461 y Ft(k)1974 4428 y FD(+)2090 4372 y(1)p 2067 4409 88 4 v 2067 4485 a(2)p Fv(k)2164 4311 y Fu(\023)2244 4428 y Fs(\001)g FD(2)2328 4353 y Fs(p)p 2396 4353 46 4 v 2396 4428 a Fv(k)26 b Fs(\024)d FD(1)523 4657 y(F)-7 b(or)27 b(the)h(basis)e(of)h (the)h(induction)e(w)n(e)i(v)n(erify)d(this)i(inequalit)n(y)d(for)k Fv(k)d FD(=)e(27:)1048 4770 y Fu(\022\022)1244 4831 y Fv(k)e Fs(\000)d FD(1)1170 4944 y(\()p Fv(k)k Fs(\000)c FD(1\))p Fv(=)p FD(2)1506 4770 y Fu(\023)1598 4831 y FD(1)p 1577 4868 83 4 v 1577 4944 a(2)1619 4920 y Ft(k)1688 4887 y FD(+)1804 4831 y(1)p 1781 4868 88 4 v 1781 4944 a(2)p Fv(k)1879 4770 y Fu(\023)1958 4887 y Fs(\001)h FD(2)2042 4812 y Fs(p)p 2110 4812 46 4 v 2110 4887 a Fv(k)26 b Fs(\031)d FD(0)p Fv(:)p FD(997755077)18 b Fv(<)23 b FD(1)p Fv(:)p Black 1922 5173 a FD(21)p Black eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.22) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(W)-7 b(e)38 b(assume)d(that)j(the)f (inequalit)n(y)d(holds)i(for)h(some)e(o)r(dd)j Fv(k)i FD(and)d(sho)n(w)f(that)i(it)e(holds)523 548 y(for)e Fv(k)26 b FD(+)d(2.)35 b(In)g(the)g(follo)n(wing)30 b(deriv)-5 b(ation)33 b(Inequalit)n(y)f(\()2361 548 y SDict begin H.S end 2361 548 a Black FD(26)p Black 2444 494 a SDict begin H.R end 2444 494 a 2444 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.26) cvn H.B /ANN pdfmark end 2444 548 a FD(\))j(follo)n(ws)d(b)n(y)i(the)i(induction)523 648 y(h)n(yp)r(othesis.)859 853 y SDict begin H.S end 859 853 a 859 853 a SDict begin 12 H.A end 859 853 a 859 853 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.26) cvn /DEST pdfmark end 859 853 a -118 x Fu(\022\022)1055 796 y Fv(k)22 b FD(+)c(2)g Fs(\000)g FD(1)981 909 y(\()p Fv(k)k FD(+)c(2)g Fs(\000)g FD(1\))p Fv(=)p FD(2)1461 735 y Fu(\023)1594 796 y FD(1)p 1532 833 167 4 v 1532 909 a(2)1574 885 y Ft(k)q FC(+2)1727 853 y FD(+)1946 796 y(1)p 1820 833 295 4 v 1820 909 a(2\()p Fv(k)j FD(+)d(2\))2125 735 y Fu(\023)2204 853 y Fs(\001)h FD(2)2288 781 y Fs(p)p 2356 781 189 4 v 2356 853 a Fv(k)j FD(+)c(2)882 1085 y(=)970 968 y Fu(\022)1111 1029 y Fv(k)s FD(\()p Fv(k)j FD(+)d(1\))p 1041 1066 439 4 v 1041 1142 a(\(\()p Fv(k)k FD(+)c(1\))p Fv(=)p FD(2\))1443 1118 y FC(2)1508 1085 y Fs(\001)1559 1029 y FD(1)p 1559 1066 42 4 v 1559 1142 a(4)1629 1085 y Fs(\001)1671 968 y Fu(\022)1806 1029 y Fv(k)j Fs(\000)d FD(1)1732 1142 y(\()p Fv(k)k Fs(\000)c FD(1\))p Fv(=)p FD(2)2068 968 y Fu(\023)2160 1029 y FD(1)p 2140 1066 83 4 v 2140 1142 a(2)2182 1118 y Ft(k)2250 1085 y FD(+)2470 1029 y(1)p 2343 1066 295 4 v 2343 1142 a(2\()p Fv(k)j FD(+)d(2\))2648 968 y Fu(\023)2728 1085 y Fs(\001)g FD(2)2811 1014 y Fs(p)p 2880 1014 189 4 v 71 x Fv(k)j FD(+)d(2)882 1317 y Fs(\024)970 1200 y Fu(\022)1113 1261 y Fv(k)p 1041 1298 V 1041 1374 a(k)k FD(+)c(1)1259 1317 y Fs(\001)1300 1200 y Fu(\022)1429 1261 y FD(1)p 1371 1298 157 4 v 1371 1386 a(2)1413 1315 y Fs(p)p 1482 1315 46 4 v 71 x Fv(k)1556 1317 y Fs(\000)1672 1261 y FD(1)p 1649 1298 88 4 v 1649 1374 a(2)p Fv(k)1746 1200 y Fu(\023)1826 1317 y FD(+)2046 1261 y(1)p 1919 1298 295 4 v 1919 1374 a(2\()p Fv(k)j FD(+)d(2\))2224 1200 y Fu(\023)2303 1317 y Fs(\001)h FD(2)2387 1246 y Fs(p)p 2456 1246 189 4 v 71 x Fv(k)i FD(+)d(2)p Black 612 w(\(26\))p Black 882 1563 a(=)980 1436 y Fs(p)p 1049 1436 46 4 v 71 x Fv(k)1095 1440 y Fs(p)p 1164 1440 189 4 v 67 x Fv(k)k FD(+)c(2)p 980 1544 374 4 v 1072 1620 a Fv(k)k FD(+)c(1)1382 1563 y Fs(\000)1475 1440 y(p)p 1544 1440 189 4 v 67 x Fv(k)j FD(+)d(2)p 1475 1544 259 4 v 1509 1620 a Fv(k)k FD(+)c(1)1761 1563 y(+)1962 1507 y(1)p 1854 1544 V 1854 1561 a Fs(p)p 1923 1561 189 4 v 67 x Fv(k)k FD(+)c(2)882 1807 y(=)980 1680 y Fs(p)p 1049 1680 46 4 v 71 x Fv(k)1095 1684 y Fs(p)p 1164 1684 189 4 v 67 x Fv(k)k FD(+)c(2)p 980 1788 374 4 v 1072 1864 a Fv(k)k FD(+)c(1)1382 1807 y(+)1475 1751 y Fv(k)j FD(+)d(1)g Fs(\000)g FD(\()p Fv(k)k FD(+)c(2\))p 1475 1788 544 4 v 1491 1872 a(\()p Fv(k)j FD(+)d(1\))1744 1805 y Fs(p)p 1813 1805 189 4 v 67 x Fv(k)k FD(+)c(2)882 2058 y(=)980 1932 y Fs(p)p 1049 1932 558 4 v 70 x Fv(k)1095 1978 y FC(2)1151 2002 y FD(+)g(2)p Fv(k)j FD(+)d(1)g Fs(\000)g FD(1)p 980 2039 628 4 v 1199 2115 a Fv(k)k FD(+)c(1)1636 2058 y Fs(\000)1964 2002 y FD(1)p 1729 2039 512 4 v 1729 2123 a(\()p Fv(k)j FD(+)d(1\))1982 2055 y Fs(p)p 2051 2055 189 4 v 68 x Fv(k)k FD(+)c(2)882 2312 y(=)980 2183 y Fu(p)p 1063 2183 434 4 v 71 x FD(\()p Fv(k)k FD(+)c(1\))1317 2230 y FC(2)1372 2254 y Fs(\000)g FD(1)p 980 2293 517 4 v 1144 2369 a Fv(k)j FD(+)d(1)1525 2312 y Fs(\000)1853 2256 y FD(1)p 1618 2293 512 4 v 1618 2377 a(\()p Fv(k)k FD(+)c(1\))1872 2309 y Fs(p)p 1941 2309 189 4 v 68 x Fv(k)j FD(+)d(2)882 2484 y Fv(<)23 b FD(1)18 b(+)g(0)882 2609 y(=)23 b(1)3350 2768 y Fs(u)-55 b(t)523 2920 y Fr(De\014nitions:)26 b FD(The)i(follo)n(wing)23 b(de\014nitions)j(are)g(used)i(in)f(Lemmata)e(\()2779 2920 y SDict begin H.S end 2779 2920 a Black FD(5)p Black 2821 2866 a SDict begin H.R end 2821 2866 a 2821 2920 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 2821 2920 a FD(\))j({)f(\()2982 2920 y SDict begin H.S end 2982 2920 a Black FD(11)p Black 3065 2866 a SDict begin H.R end 3065 2866 a 3065 2920 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.11) cvn H.B /ANN pdfmark end 3065 2920 a FD(\).)p Black 575 3075 a Fr({)p Black 41 w FD(Let)i Fv(S)870 3045 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1097 3075 y FD(=)24 b Fs(f)p Fv(y)1269 3087 y FC(1)1331 3075 y Fv(>)h(y)1462 3087 y FC(2)1524 3075 y Fv(>)f Fs(\001)14 b(\001)g(\001)25 b Fv(>)g(y)1866 3087 y Ft(m)1928 3075 y Fs(g)g(\022)g Fv(S)33 b FD(b)r(e)c(random)e(v) -5 b(ariables)26 b(for)i(the)h(set)g(of)664 3175 y(items)d(arriving)e (during)i(the)i(time)e(p)r(erio)r(d)h([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\),)p Black 575 3270 a Fr({)p Black 41 w FD(and)26 b(let)g Fv(S)999 3240 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1224 3270 y FD(=)d Fs(f)p Fv(z)1393 3282 y FC(1)1452 3270 y Fv(>)g(z)1579 3282 y FC(2)1639 3270 y Fv(>)f Fs(\001)14 b(\001)g(\001)23 b Fv(>)g(z)1973 3285 y Fq(j)o Ft(S)s Fq(j\000)p Ft(m)2171 3270 y Fs(g)f(\022)h Fv(S)31 b FD(b)r(e)c(random)d(v)-5 b(ariables)24 b(for)i(the)664 3369 y(set)i(of)f(items)f(arriving)e(during)i(the)i(time)e(p)r(erio)r (d)h([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\).)p Black 575 3464 a Fr({)p Black 41 w FD(Let)21 b Fv(Q)i FD(=)982 3394 y Fu(\014)982 3444 y(\014)1010 3464 y Fs(f)p Fv(z)j Fs(2)d Fv(S)1251 3434 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1476 3464 y Fs(j)g Fv(z)j(>)d(T)12 b Fs(g)1777 3394 y Fu(\014)1777 3444 y(\014)1825 3464 y FD(b)r(e)21 b(the)g(n)n(um)n(b)r(er)e(of)i(elemen)n(ts)e(in)g Fv(S)2928 3434 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3151 3464 y FD(greater)664 3564 y(than)31 b(the)g(threshold)f Fv(T)12 b FD(.)30 b(In)h(the)g(Algorithm)2146 3564 y SDict begin H.S end 2146 3564 a Black FD(1)p Black 2187 3511 a SDict begin H.R end 2187 3511 a 2187 3564 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (algocfline.1) cvn H.B /ANN pdfmark end 2187 3564 a 31 w FD(w)n(e)g(de\014ne)g(the)g(threshold)f Fv(T)42 b FD(where)664 3664 y Fv(T)h FD(=)31 b Fv(y)893 3676 y Ft(\034)935 3664 y FD(,)h Fv(\034)42 b FD(=)31 b Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(,)i(if)1504 3593 y Fu(\014)1504 3643 y(\014)1532 3664 y Fv(S)1588 3634 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1789 3593 y Fu(\014)1789 3643 y(\014)1849 3664 y Fs(\025)e Fv(\034)43 b FD(and)32 b Fv(T)43 b FD(=)32 b(0)g(otherwise)f(\(in)h(this)g(case)g(w)n(e)664 3763 y(consider)26 b(an)n(y)h(item)f(of)h(v)-5 b(alue)27 b(0)g(as)g(greater)f(than)h Fv(T)12 b FD(\).)p Black 575 3858 a Fr({)p Black 41 w FD(Let)36 b Fv(G)886 3870 y Ft(i)914 3858 y FD(,)g(1)g Fs(\024)g Fv(i)h Fs(\024)f Fv(\034)9 b FD(,)36 b(b)r(e)h(iden)n(tical)32 b(indep)r(enden)n(t)k (geometric)c(random)i(v)-5 b(ariables,)664 3971 y(suc)n(h)37 b(that)h(for)f(an)n(y)g(in)n(teger)f Fv(j)44 b Fs(\025)39 b FD(0,)f(Prob)n([)14 b Fv(G)2203 3983 y Ft(i)2254 3971 y FD(=)23 b Fv(j)c FD(])39 b(=)2561 3904 y Fu(\000)2609 3939 y FC(1)p 2609 3953 34 4 v 2609 4000 a(2)2653 3904 y Fu(\001)2691 3921 y Ft(j)s FC(+1)2810 3971 y FD(.)e(It)h(follo)n(ws)d (that)664 4071 y(the)40 b(exp)r(ectation)e(of)h Fv(G)1449 4083 y Ft(i)1477 4071 y FD(,)g(E[)o Fv(G)1683 4083 y Ft(i)1711 4071 y FD(])k(=)e(1,)e(whereas)f(the)h(v)-5 b(ariance)37 b Fv(\033)2854 4041 y FC(2)2892 4071 y FD([)p Fv(G)2980 4083 y Ft(i)3008 4071 y FD(])42 b(=)g(2.)d(Let)664 4171 y Fv(G)24 b FD(=)840 4108 y Fu(P)928 4129 y Ft(\034)928 4195 y(i)p FC(=1)1053 4171 y Fv(G)1118 4183 y Ft(i)1146 4171 y FD(.)k(Note)g(also)d(that)j(E[)p Fv(G)p FD(])23 b(=)g Fv(\034)37 b FD(and)27 b Fv(\033)2307 4140 y FC(2)2345 4171 y FD([)p Fv(G)p FD(])d(=)e(2)p Fv(\034)9 b FD(.)p Black 575 4266 a Fr({)p Black 41 w FD(W)-7 b(e)25 b(abbreviate)d(E)1266 4281 y Ft(\022)r FC(:)p Ft(S)s Fq(7!)p FC([0)p Ft(;)p FC(1\))1563 4266 y FD([)p Fv(v)s FD(\()p Fv(C)1720 4278 y Ft(k)q(;\015)1820 4266 y FD(\()p Fv(S;)14 b(\022)r FD(\)\)])25 b(as)f(E[)o Fv(C)2330 4278 y Ft(k)q(;\015)2430 4266 y FD(\()p Fv(S)5 b FD(\)])24 b(as)g(in)g(the)g(pro)r(of)g(of)g (The-)664 4365 y(orem)872 4365 y SDict begin H.S end 872 4365 a Black FD(2)p Black 913 4312 a SDict begin H.R end 913 4312 a 913 4365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.2) cvn H.B /ANN pdfmark end 913 4365 a FD(.)523 4425 y SDict begin H.S end 523 4425 a 523 4425 a SDict begin 12 H.A end 523 4425 a 523 4425 a SDict begin [/View [/XYZ H.V]/Dest (lemma.5) cvn /DEST pdfmark end 523 4425 a Black 100 x Fr(Lemma)30 b(5.)p Black 40 w Fp(Given)43 b(that)f Fs(j)q Fv(S)5 b Fs(j)45 b(\025)g FD(3)p Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)o Fp(,)e(for)g(any)g Fv(\015)50 b Fs(\024)45 b Fv(\015)2552 4495 y Fq(\003)2636 4525 y FD(=)g(0)p Fv(:)p FD(003176)39 b Fp(and)k(any)523 4625 y Fv(k)26 b Fs(\024)d FD(1)p Fv(=\015)5 b Fp(,)28 b(the)i(exp)l(e)l (cte)l(d)g(pr)l(o\014t)g(of)g Fv(C)1700 4637 y Ft(k)q(;\015)1829 4625 y Fp(during)g(time)g(interval)h FD([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))27 b Fp(is)k(at)e(le)l(ast)532 4862 y FD(E)589 4770 y Fu(h)628 4862 y Fv(C)693 4819 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))687 4887 y Ft(k)q(;\015)895 4862 y FD(\()p Fv(S)5 b FD(\))1015 4770 y Fu(i)1078 4862 y Fs(\025)1273 4806 y FD(1)p 1175 4843 237 4 v 1175 4919 a(1)18 b(+)g Fv(\015)5 b(k)1422 4770 y Fu(\020)1471 4862 y FD(1)18 b Fs(\000)1614 4789 y(p)p 1684 4789 42 4 v 1684 4862 a FD(2)f Fs(\001)i Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2207 4770 y Fu(\021)2255 4720 y( )2321 4862 y FD(1)g Fs(\000)2474 4804 y FD(2)2516 4733 y Fu(p)p 2599 4733 272 4 v 71 x FD(1)f(+)h(1)p Fv(=k)p 2474 4843 397 4 v 2615 4860 a Fs(p)p 2684 4860 46 4 v 70 x Fv(k)2899 4862 y Fs(\000)2994 4806 y FD(1)p 2992 4843 V 2992 4919 a Fv(k)3048 4720 y Fu(!)3123 4806 y Fv(T)3172 4818 y Ft(k)3213 4806 y FD(\()p Fv(S)5 b FD(\))p 3123 4843 210 4 v 3207 4919 a(2)3373 4862 y Fv(;)p Black 1922 5173 a FD(22)p Black eop end end %%Page: 23 23 TeXDict begin HPSdict begin 23 22 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.23) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a Fp(wher)l(e)30 b Fv(\013)p FD(\()p Fv(\015)5 b FD(\))24 b(=)e(3)1200 377 y Fu(p)p 1283 377 326 4 v 71 x Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))29 b Fp(as)h(de\014ne)l(d)g(in)g(The)l(or)l(em)2471 448 y SDict begin H.S end 2471 448 a Black Fp(3)p Black 2514 395 a SDict begin H.R end 2514 395 a 2514 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2514 448 a Fp(.)p Black 523 649 a(Pr)l(o)l(of.)p Black 43 w FD(Splitting)25 b(the)j(probabilit)n(y)c(space)j(b)n(y)g (conditioning)d(on)k Fv(Q)f FD(w)n(e)g(obtain)849 906 y SDict begin H.S end 849 906 a 849 906 a SDict begin 12 H.A end 849 906 a 849 906 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.27) cvn /DEST pdfmark end 849 906 a FD(E)906 814 y Fu(h)944 906 y Fv(C)1009 863 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1003 931 y Ft(k)q(;\015)1212 906 y FD(\()p Fv(S)5 b FD(\))1332 814 y Fu(i)1394 906 y FD(=)1509 803 y Fq(1)1482 827 y Fu(X)1484 1003 y Ft(q)r FC(=0)1616 906 y FD(Prob)o([)13 b Fv(Q)23 b FD(=)g Fv(q)17 b FD(])h Fs(\001)h FD(E)2200 814 y Fu(h)2238 906 y Fv(C)2303 863 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2297 931 y Ft(k)q(;\015)2506 906 y FD(\()p Fv(S)5 b FD(\))23 b Fs(j)g Fv(Q)g FD(=)f Fv(q)2911 814 y Fu(i)1394 1219 y Fs(\025)1482 1108 y FC(2)p Fq(d)q Ft(k)q(=)p FC(2)q Fq(e)1521 1140 y Fu(X)1523 1316 y Ft(q)r FC(=0)1694 1219 y FD(Prob)n([)14 b Fv(Q)23 b FD(=)g Fv(q)17 b FD(])h Fs(\001)h FD(E)2278 1127 y Fu(h)2316 1219 y Fv(C)2381 1176 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2375 1244 y Ft(k)q(;\015)2583 1219 y FD(\()p Fv(S)5 b FD(\))24 b Fs(j)f Fv(Q)g FD(=)f Fv(q)2989 1127 y Fu(i)3056 1219 y Fv(:)p Black 178 w FD(\(27\))p Black 648 1542 a(Lemma)947 1542 y SDict begin H.S end 947 1542 a Black FD(6)p Black 988 1488 a SDict begin H.R end 988 1488 a 988 1542 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.6) cvn H.B /ANN pdfmark end 988 1542 a 31 w FD(sho)n(ws)29 b(that)i(E)1500 1450 y Fu(h)1539 1542 y Fv(C)1604 1499 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1598 1567 y Ft(k)q(;\015)1806 1542 y FD(\()p Fv(S)5 b FD(\))24 b Fs(j)f Fv(Q)f FD(=)h Fv(q)2212 1450 y Fu(i)2279 1542 y Fs(\025)2445 1509 y FC(1)p 2382 1523 160 4 v 2382 1570 a(1+)p Ft(\015)t(k)2551 1475 y Fu(\000)2589 1542 y FD(1)18 b Fs(\000)2732 1473 y(p)p 2801 1473 42 4 v 69 x FD(2)g Fs(\001)h Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)3325 1475 y Fu(\001)3382 1542 y Fs(\001)523 1607 y Fu(\020)573 1700 y FD(1)g Fs(\000)g FD(2)768 1659 y Fq(j)p Ft(q)r Fq(\000d)p Ft(k)q(=)p FC(2)p Fq(e)q(j)p 767 1680 289 4 v 893 1728 a Ft(k)1084 1700 y Fs(\000)1179 1667 y FC(1)p 1177 1681 37 4 v 1177 1728 a Ft(k)1224 1607 y Fu(\021)1292 1700 y Fs(\001)1343 1659 y Ft(T)1382 1668 y Fo(k)1419 1659 y FC(\()p Ft(S)s FC(\))p 1343 1680 172 4 v 1412 1728 a(2)1524 1700 y FD(.)28 b(Substituting)f(this)g(lo)n (w)n(er)e(b)r(ound)j(in)e(\()2814 1700 y SDict begin H.S end 2814 1700 a Black FD(27)p Black 2898 1646 a SDict begin H.R end 2898 1646 a 2898 1700 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.27) cvn H.B /ANN pdfmark end 2898 1700 a FD(\))i(w)n(e)f(obtain)660 2055 y(E)717 1963 y Fu(h)755 2055 y Fv(C)820 2012 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))814 2080 y Ft(k)q(;\015)1023 2055 y FD(\()p Fv(S)5 b FD(\))1143 1963 y Fu(i)1205 2055 y Fs(\025)1293 1993 y Fu(P)1380 2014 y FC(2)p Fq(d)q Ft(k)q(=)p FC(2)q Fq(e)1380 2080 y Ft(q)r FC(=0)1596 1913 y Fu( )1662 2055 y FD(Prob)o([)13 b Fv(Q)23 b FD(=)g Fv(q)17 b FD(])h Fs(\001)2189 1938 y Fu(\024)2306 2023 y FC(1)p 2242 2037 160 4 v 2242 2084 a(1+)p Ft(\015)t(k)2412 1938 y Fu(\022)2473 2055 y FD(1)g Fs(\000)2616 1987 y(p)p 2685 1987 42 4 v 68 x FD(2)g Fs(\001)g Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)3208 1938 y Fu(\023)1250 2304 y Fs(\001)1273 2187 y Fu(\022)1334 2304 y FD(1)g Fs(\000)g FD(2)1529 2264 y Fq(j)p Ft(q)r Fq(\000d)p Ft(k)q(=)p FC(2)q Fq(e)q(j)p 1529 2285 289 4 v 1655 2333 a Ft(k)1846 2304 y Fs(\000)1941 2272 y FC(1)p 1939 2286 37 4 v 1939 2333 a Ft(k)1985 2187 y Fu(\023)2065 2304 y Fs(\001)2117 2264 y Ft(T)2156 2273 y Fo(k)2192 2264 y FC(\()p Ft(S)s FC(\))p 2117 2285 172 4 v 2186 2333 a(2)2298 2187 y Fu(\025)2342 2162 y(!)1132 2492 y FD(=)1293 2459 y FC(1)p 1230 2473 160 4 v 1230 2521 a(1+)p Ft(\015)t(k)1417 2492 y Fs(\001)1459 2425 y Fu(\000)1497 2492 y FD(1)g Fs(\000)1640 2423 y(p)p 1709 2423 42 4 v 69 x FD(2)g Fs(\001)h Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2233 2425 y Fu(\001)2279 2452 y Ft(T)2318 2461 y Fo(k)2355 2452 y FC(\()p Ft(S)s FC(\))p 2279 2473 172 4 v 2348 2521 a(2)1250 2627 y Fs(\001)1287 2565 y Fu(P)1375 2586 y FC(2)p Fq(d)p Ft(k)q(=)p FC(2)q Fq(e)1375 2652 y Ft(q)r FC(=0)1591 2627 y FD(Prob)n([)c Fv(Q)23 b FD(=)f Fv(q)17 b FD(])2057 2535 y Fu(\020)2107 2627 y FD(1)h Fs(\000)g FD(2)2302 2587 y Fq(j)p Ft(q)r Fq(\000d)p Ft(k)q(=)p FC(2)q Fq(ej)p 2301 2608 289 4 v 2427 2656 a Ft(k)2618 2627 y Fs(\000)2713 2595 y FC(1)p 2711 2609 37 4 v 2711 2656 a Ft(k)2758 2535 y Fu(\021)648 2854 y FD(Lemma)940 2854 y SDict begin H.S end 940 2854 a Black FD(7)p Black 982 2801 a SDict begin H.R end 982 2801 a 982 2854 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.7) cvn H.B /ANN pdfmark end 982 2854 a 24 w FD(pro)n(v)n(es)k(that)i(if)f Fs(j)p Fv(S)5 b Fs(j)23 b(\025)g FD(3)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)h FD(then)g(for)g(an)n(y)f Fv(q)j Fs(\024)d FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)h FD(the)g(probabilit)n(y)523 2954 y(that)42 b Fv(Q)j FD(=)h Fv(q)e FD(is)c(the)i(same)e(as)g(the)i (probabilit)n(y)37 b(that)42 b(the)f(sum)g(of)g Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)g FD(iden)n(tical)523 3054 y(geometrical)28 b(random)i(v)-5 b(ariables)29 b(is)i(equal)g Fv(q)s FD(.)i(W)-7 b(e)33 b(denote)f(this)f(sum)h(b)n(y)g Fv(G)h FD(and)f(deriv)n(e)523 3153 y(the)c(follo)n(wing)c(lo)n(w)n(er)h(b)r(ound)j(on)f(the)h(last)e (sum)h(of)g(Equation)f(\()2566 3153 y SDict begin H.S end 2566 3153 a Black FD(15)p Black 2649 3100 a SDict begin H.R end 2649 3100 a 2649 3153 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.27) cvn H.B /ANN pdfmark end 2649 3153 a FD(\):)1081 3527 y SDict begin H.S end 1081 3527 a 1081 3527 a SDict begin 12 H.A end 1081 3527 a 1081 3527 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.28) cvn /DEST pdfmark end 1081 3527 a -111 x FC(2)p Fq(d)q Ft(k)q(=)p FC(2)p Fq(e)1120 3448 y Fu(X)1122 3624 y Ft(q)r FC(=0)1293 3527 y FD(Prob)n([)14 b Fv(Q)23 b FD(=)g Fv(q)17 b FD(])1760 3410 y Fu(\022)1821 3527 y FD(1)h Fs(\000)g FD(2)2016 3470 y Fs(j)p Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)n Fs(ej)p 2016 3508 391 4 v 2187 3584 a Fv(k)2434 3527 y Fs(\000)2529 3470 y FD(1)p 2527 3508 46 4 v 2527 3584 a Fv(k)2583 3410 y Fu(\023)1104 3839 y FD(=)1192 3728 y FC(2)p Fq(d)q Ft(k)q(=)p FC(2)q Fq(e)1231 3760 y Fu(X)1233 3936 y Ft(q)r FC(=0)1404 3839 y FD(Prob)n([)c Fv(G)24 b FD(=)e Fv(q)17 b FD(])1870 3722 y Fu(\022)1931 3839 y FD(1)h Fs(\000)g FD(2)2126 3783 y Fs(j)p Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)o Fs(ej)p 2126 3820 391 4 v 2298 3896 a Fv(k)2545 3839 y Fs(\000)2640 3783 y FD(1)p 2638 3820 46 4 v 2638 3896 a Fv(k)2693 3722 y Fu(\023)1104 4152 y FD(=)1192 4041 y Fq(d)q Ft(k)q(=)p FC(2)p Fq(e)1215 4073 y Fu(X)1216 4249 y Ft(r)r FC(=0)1371 4152 y FD(Prob)n([)c Fs(j)p Fv(G)19 b Fs(\000)f(d)p Fv(k)s(=)p FD(2)o Fs(ej)23 b FD(=)g Fv(r)16 b FD(])j Fs(\001)2247 4035 y Fu(\022)2308 4152 y FD(1)f Fs(\000)g FD(2)g Fs(\001)2566 4096 y Fv(r)p 2563 4133 V 2563 4209 a(k)2637 4152 y Fs(\000)2732 4096 y FD(1)p 2730 4133 V 2730 4209 a Fv(k)2786 4035 y Fu(\023)p Black 3257 4152 a FD(\(28\))p Black 1104 4427 a Fs(\025)1219 4324 y Fq(1)1192 4349 y Fu(X)1194 4524 y Ft(r)r FC(=0)1326 4427 y FD(Prob)o([)c Fs(j)p Fv(G)k Fs(\000)g(d)p Fv(k)s(=)p FD(2)o Fs(ej)23 b FD(=)g Fv(r)16 b FD(])j Fs(\001)2202 4310 y Fu(\022)2263 4427 y FD(1)f Fs(\000)g FD(2)g Fs(\001)2521 4371 y Fv(r)p 2518 4408 V 2518 4484 a(k)2592 4427 y Fs(\000)2687 4371 y FD(1)p 2685 4408 V 2685 4484 a Fv(k)2741 4310 y Fu(\023)p Black 3257 4427 a FD(\(29\))p Black 1104 4700 a(=)23 b(1)18 b Fs(\000)g FD(2)g Fs(\001)1447 4639 y FD(E)1504 4572 y Fu(\002)1538 4639 y Fs(j)p Fv(G)h Fs(\000)1728 4572 y Fu(\006)1777 4606 y Ft(k)p 1777 4620 37 4 v 1779 4668 a FC(2)1823 4572 y Fu(\007)1863 4639 y Fs(j)1886 4572 y Fu(\003)p 1447 4681 474 4 v 1661 4757 a Fv(k)1949 4700 y Fs(\000)2044 4644 y FD(1)p 2042 4681 46 4 v 2042 4757 a Fv(k)1104 4909 y FD(=)k(1)18 b Fs(\000)g FD(2)g Fs(\001)1447 4853 y FD(E[)o Fs(j)p Fv(G)h Fs(\000)f FD(E[)p Fv(G)p FD(])p Fs(j)p FD(])p 1447 4890 484 4 v 1665 4966 a Fv(k)1958 4909 y Fs(\000)2054 4853 y FD(1)p 2051 4890 46 4 v 2051 4966 a Fv(k)2163 4909 y(:)p Black 1071 w FD(\(30\))p Black Black 1922 5173 a(23)p Black eop end end %%Page: 24 24 TeXDict begin HPSdict begin 24 23 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.24) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(Equalit)n(y)24 b(\()890 448 y SDict begin H.S end 890 448 a Black FD(28)p Black 974 395 a SDict begin H.R end 974 395 a 974 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.28) cvn H.B /ANN pdfmark end 974 448 a FD(\))j(follo)n(ws)e(b)n(y)i(the)g(c)n(hange)f(of)i(v) -5 b(ariables)24 b Fv(r)h FD(=)e Fs(j)p Fv(q)e Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(ej)p FD(,)28 b(Inequalit)n(y)c(\()3289 448 y SDict begin H.S end 3289 448 a Black FD(29)p Black 3373 395 a SDict begin H.R end 3373 395 a 3373 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.28) cvn H.B /ANN pdfmark end 3373 448 a FD(\))523 548 y(follo)n(ws)h(since)999 481 y Fu(\000)1038 548 y FD(1)17 b Fs(\000)h FD(2)g Fs(\001)1294 515 y Ft(r)p 1292 529 37 4 v 1292 577 a(k)1357 548 y Fs(\000)1452 515 y FC(1)p 1450 529 V 1450 577 a Ft(k)1497 481 y Fu(\001)1558 548 y Fv(<)23 b FD(0)28 b(for)f(an)n(y)g Fv(r)f(>)d Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)28 b FD(and)f(the)h(last)f (equalit)n(y)e(follo)n(ws)523 648 y(since)h Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)d FD(=)g Fv(E)5 b FD(\()p Fv(G)p FD(\).)648 756 y(Using)18 b(the)j(fact)f(that)g(E[)p Fs(j)p Fv(W)30 b Fs(\000)18 b FD(E[)p Fv(W)12 b FD(])p Fs(j)p FD(])23 b Fs(\024)1981 685 y Fu(p)p 2064 685 266 4 v 71 x FD(V)-7 b(ar)o([)p Fv(W)12 b FD(])20 b(\(V)-7 b(ar)20 b(here)g(stands)f(for)h (v)-5 b(ariance\))523 856 y(for)20 b(an)n(y)f(random)g(v)-5 b(ariable)17 b Fv(W)32 b FD(\(Lemma)1826 856 y SDict begin H.S end 1826 856 a Black FD(8)p Black 1867 802 a SDict begin H.R end 1867 802 a 1867 856 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.8) cvn H.B /ANN pdfmark end 1867 856 a FD(\))21 b(and)f(the)h(fact)f(that)h(V)-7 b(ar)o([)p Fv(G)p FD(])24 b(=)e(2)p Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)h(\024)f Fv(k)7 b FD(+)t(1)523 956 y(w)n(e)27 b(get)1264 1142 y SDict begin H.S end 1264 1142 a 1264 1142 a SDict begin 12 H.A end 1264 1142 a 1264 1142 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.31) cvn /DEST pdfmark end 1264 1142 a 1274 1086 a FD(E[)o Fs(j)p Fv(G)19 b Fs(\000)f FD(E[)p Fv(G)p FD(])p Fs(j)p FD(])p 1274 1123 484 4 v 1492 1199 a Fv(k)1790 1142 y Fs(\024)1888 1019 y(p)p 1957 1019 189 4 v 67 x Fv(k)j FD(+)d(1)p 1888 1123 259 4 v 1994 1199 a Fv(k)2179 1142 y FD(=)2276 1014 y Fu(p)p 2359 1014 272 4 v 71 x FD(1)g(+)g(1)p Fv(=k)p 2276 1123 355 4 v 2396 1140 a Fs(p)p 2465 1140 46 4 v 71 x Fv(k)2641 1142 y(:)593 b FD(\(31\))648 1376 y(Substituting)23 b(the)i(upp)r(er)g(b)r(ound)g(of)g(Equation)e(\()2228 1376 y SDict begin H.S end 2228 1376 a Black FD(31)p Black 2311 1323 a SDict begin H.R end 2311 1323 a 2311 1376 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.31) cvn H.B /ANN pdfmark end 2311 1376 a FD(\))i(in)n(to)f(Equation)e(\()2921 1376 y SDict begin H.S end 2921 1376 a Black FD(30)p Black 3005 1323 a SDict begin H.R end 3005 1323 a 3005 1376 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.28) cvn H.B /ANN pdfmark end 3005 1376 a FD(\),)j(and)f(then)523 1476 y(substituting)i(the)i(resulting)d(inequalit)n(y)f(in)n(to)j (Equation)e(\()2414 1476 y SDict begin H.S end 2414 1476 a Black FD(15)p Black 2498 1422 a SDict begin H.R end 2498 1422 a 2498 1476 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.27) cvn H.B /ANN pdfmark end 2498 1476 a FD(\))j(w)n(e)f(obtain)f(the)i(lemma.)3350 1585 y Fs(u)-55 b(t)648 1795 y FD(It)22 b(ma)n(y)e(b)r(e)i(useful)f(to)h(observ)n(e)e (that)i(lo)n(w)n(er)d(b)r(ound)j(giv)n(en)e(in)h(the)h(follo)n(wing)c (Lemma)3340 1795 y SDict begin H.S end 3340 1795 a Black FD(6)p Black 3382 1742 a SDict begin H.R end 3382 1742 a 3382 1795 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.6) cvn H.B /ANN pdfmark end 3382 1795 a FD(,)523 1895 y(is)j(appro)n(ximately)d(equal)j(to)h(the)h(exp)r(ected)g(size)e(of)h (the)h Fv(\015)5 b FD(-indep)r(enden)n(t)21 b(set)i(c)n(hosen)e(from) 523 1994 y(the)26 b Fv(q)j FD(items)24 b(arriving)e(during)i(the)i (time)e(in)n(terv)-5 b(al)23 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))25 b(divided)f(b)n(y)h Fv(q)s FD(,)h(m)n(ultiplied)21 b(b)n(y)523 2119 y(the)28 b(exp)r(ected)g(v)-5 b(alue)26 b(of)i(these)g Fv(q)i FD(elemen)n(ts)c(\()1973 2027 y Fu(\020)2023 2119 y FD(1)18 b Fs(\000)g FD(2)2218 2079 y Fq(j)p Ft(q)r Fq(\000d)p Ft(k)q(=)p FC(2)p Fq(e)q(j)p 2217 2100 289 4 v 2343 2148 a Ft(k)2534 2119 y Fs(\000)2629 2086 y FC(1)p 2627 2100 37 4 v 2627 2148 a Ft(k)2674 2027 y Fu(\021)2742 2119 y Fs(\001)2793 2079 y Ft(T)2832 2088 y Fo(k)2869 2079 y FC(\()p Ft(S)s FC(\))p 2793 2100 172 4 v 2862 2148 a(2)2975 2119 y FD(\).)523 2173 y SDict begin H.S end 523 2173 a 523 2173 a SDict begin 12 H.A end 523 2173 a 523 2173 a SDict begin [/View [/XYZ H.V]/Dest (lemma.6) cvn /DEST pdfmark end 523 2173 a Black 190 x Fr(Lemma)30 b(6.)p Black 40 w Fp(F)-6 b(or)32 b(any)g Fv(\015)f Fs(\024)26 b Fv(\015)1508 2333 y Fq(\003)1571 2363 y FD(=)g(0)p Fv(:)p FD(003176)p Fp(,)j(any)j Fv(k)d Fs(\024)d FD(1)p Fv(=\015)5 b Fp(,)30 b(any)i Fv(S)5 b Fp(,)31 b Fs(j)p Fv(S)5 b Fs(j)27 b(\025)e Fv(k)s Fp(,)32 b(and)g(for)523 2463 y(any)e FD(0)23 b Fs(\024)f Fv(q)k Fs(\024)d FD(2)p Fs(d)p Fv(k)s(=)p FD(2)n Fs(e)30 b Fp(we)g(have)754 2689 y FD(E)811 2597 y Fu(h)850 2689 y Fv(C)915 2646 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))909 2714 y Ft(k)q(;\015)1117 2689 y FD(\()p Fv(S)5 b FD(\))23 b Fs(j)g Fv(Q)g FD(=)g Fv(q)1523 2597 y Fu(i)800 2896 y Fs(\025)995 2840 y FD(1)p 898 2877 237 4 v 898 2953 a(1)18 b(+)g Fv(\015)5 b(k)1144 2804 y Fu(\020)1194 2896 y FD(1)18 b Fs(\000)1337 2824 y(p)p 1406 2824 42 4 v 72 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)1870 2804 y Fu(\021)1918 2779 y(\022)1979 2896 y FD(1)g Fs(\000)g FD(2)g Fs(\001)2233 2840 y(j)p Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(ej)p 2233 2877 391 4 v 2406 2953 a Fv(k)2652 2896 y Fs(\000)2747 2840 y FD(1)p 2745 2877 46 4 v 2745 2953 a Fv(k)2801 2779 y Fu(\023)2872 2840 y Fv(T)2921 2852 y Ft(k)2961 2840 y FD(\()p Fv(S)5 b FD(\))p 2872 2877 210 4 v 2956 2953 a(2)3151 2896 y Fv(:)p Black 523 3167 a Fp(Pr)l(o)l(of.)p Black 43 w FD(W)-7 b(e)21 b(condition)d(up)r(on)i(there)g(b)r(eing)f Fv(q)k FD(items)18 b(exceeding)h(the)h(threshold)f(in)g Fv(S)3180 3137 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3382 3167 y FD(.)523 3267 y(Let)28 b Fv(S)723 3279 y Ft(>T)855 3267 y FD(b)r(e)g(the)g(set)f(of)h(all)d(items)h(in)h Fv(S)32 b FD(greater)26 b(than)i Fv(T)12 b FD(.)648 3376 y(Assume)27 b Fs(j)p Fv(S)1035 3346 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1237 3376 y Fs(j)d(\025)g Fv(\034)38 b FD(\()p Fw(i.e.)p FD(,)27 b(su\016cien)n(tly)g(man)n(y)f(items)h (arriv)n(e)e(b)r(efore)j(time)f(1)p Fv(=)p FD(2)g(so)523 3475 y(as)32 b(to)g(tak)n(e)f Fv(y)961 3487 y Ft(\034)1035 3475 y FD(as)g(a)h(threshold\).)f(Therefore)g(there)h(are)g(exactly)e Fv(\034)h Fs(\000)22 b FD(1)31 b(items)g Fv(>)f(y)3262 3487 y Ft(\034)3336 3475 y FD(in)523 3575 y Fv(S)579 3545 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))781 3575 y FD(,)d(and)g(from)f(the)h(conditioning)d(there)j(are)f(exactly)g Fv(q)k FD(items)25 b Fv(>)e(y)2858 3587 y Ft(\034)2926 3575 y FD(in)k Fv(S)3079 3545 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3280 3575 y FD(,)h(so)523 3675 y Fs(j)p Fv(S)597 3687 y Ft(>T)701 3675 y Fs(j)j FD(=)g Fv(\034)h Fs(\000)21 b FD(1)g(+)g Fv(q)36 b FD(is)31 b(the)i(n)n(um)n(b)r(er)e (items)g(in)h Fv(S)37 b FD(that)c(are)e(strictly)f(greater)h(than)i (the)523 3774 y(threshold)26 b Fv(y)929 3786 y Ft(\034)970 3774 y FD(.)i(Th)n(us,)f(the)h(probabilit)n(y)c(that)k(an)f(item)f Fv(x)e Fs(2)f Fv(S)2506 3786 y Ft(>T)2638 3774 y FD(is)j(in)h Fv(S)2874 3744 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3104 3774 y FD(is)1055 3996 y(Prob)1232 3904 y Fu(h)1285 3996 y Fv(x)c Fs(2)h Fv(S)1490 3962 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1714 3996 y Fs(j)f Fv(Q)g FD(=)g Fv(q)1990 3904 y Fu(i)2053 3996 y FD(=)2231 3940 y Fv(q)p 2150 3977 202 4 v 2150 4053 a Fs(j)p Fv(S)2224 4065 y Ft(>T)2329 4053 y Fs(j)2385 3996 y FD(=)2627 3940 y Fv(q)p 2482 3977 331 4 v 2482 4053 a(\034)28 b FD(+)18 b Fv(q)k Fs(\000)c FD(1)2850 3996 y Fv(:)648 4266 y FD(If)28 b Fs(j)p Fv(S)810 4236 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))1012 4266 y Fs(j)d Fv(<)f(\034)9 b FD(,)29 b(the)g(threshold)e(is)g(set)h(to)h(b) r(e)g(zero,)e(and)h Fv(S)h FD(=)c Fv(S)2769 4278 y Ft(>T)2873 4266 y FD(.)j(The)h(size)e(of)h Fv(S)523 4366 y FD(is)g Fs(j)p Fv(S)687 4335 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))888 4366 y Fs(j)20 b FD(+)e Fv(q)28 b(<)d(\034)k FD(+)18 b Fv(q)s FD(.)29 b(It)g(follo)n(ws)d(that)j(the)g(probabilit)n(y)c (that)k(an)f(item)g Fv(x)d Fs(2)g Fv(S)3216 4378 y Ft(>T)3349 4366 y FD(is)523 4465 y(in)i Fv(S)676 4435 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))905 4465 y FD(is)1055 4666 y(Prob)1232 4574 y Fu(h)1285 4666 y Fv(x)c Fs(2)h Fv(S)1490 4632 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1714 4666 y Fs(j)f Fv(Q)g FD(=)g Fv(q)1990 4574 y Fu(i)2053 4666 y FD(=)2231 4610 y Fv(q)p 2150 4647 202 4 v 2150 4723 a Fs(j)p Fv(S)2224 4735 y Ft(>T)2329 4723 y Fs(j)2385 4666 y(\025)2627 4610 y Fv(q)p 2482 4647 331 4 v 2482 4723 a(\034)28 b FD(+)18 b Fv(q)k Fs(\000)c FD(1)2850 4666 y Fv(:)648 4924 y FD(W)-7 b(e)28 b(no)n(w)f(consider)e(t)n(w)n(o)i (cases:)f Fv(q)h Fs(\024)22 b(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(,)28 b(and)f Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)c Fv(<)f(q)k Fs(\024)d FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)p FD(.)p Black 1922 5173 a(24)p Black eop end end %%Page: 25 25 TeXDict begin HPSdict begin 25 24 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.25) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(F)-7 b(or)24 b Fv(q)i Fs(\024)c(d)p Fv(k)s(=)p FD(2)o Fs(e)j FD(the)g(exp)r(ected)g(sum)e(of)i(the)g(v)-5 b(alues)23 b(of)i(the)g Fv(q)i FD(items)c(in)h Fv(S)2968 418 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))3182 448 y Fs(\\)12 b Fv(S)3300 460 y Ft(>T)523 548 y FD(is)523 791 y SDict begin H.S end 523 791 a 523 791 a SDict begin 12 H.A end 523 791 a 523 791 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.32) cvn /DEST pdfmark end 523 791 a FD(E)580 699 y Fu(h)619 791 y Fv(v)s FD(\()p Fs(f)p Fv(x)23 b Fs(j)g Fv(x)h Fs(2)f Fv(S)1057 757 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1277 791 y Fs(\\)c Fv(S)1402 803 y Ft(>T)1506 791 y Fs(g)p FD(\))k Fs(j)g Fv(Q)g FD(=)g Fv(q)1866 699 y Fu(i)1928 791 y FD(=)2063 713 y Fu(X)2016 891 y Ft(x)p Fq(2)p Ft(S)2140 899 y Fo(>T)2244 791 y FD(Prob)2420 699 y Fu(h)2473 791 y Fv(x)h Fs(2)f Fv(S)2678 757 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2903 791 y Fs(j)g Fv(Q)g FD(=)f Fv(q)3179 699 y Fu(i)3237 791 y Fs(\001)d Fv(v)s FD(\()p Fv(x)p FD(\))1928 1037 y(=)2106 980 y Fv(q)p 2026 1018 202 4 v 2026 1094 a Fs(j)p Fv(S)2100 1106 y Ft(>T)2204 1094 y Fs(j)2255 1037 y(\001)2344 958 y Fu(X)2297 1136 y Ft(x)p Fq(2)p Ft(S)2421 1144 y Fo(>T)2525 1037 y Fv(v)s FD(\()p Fv(x)p FD(\))1928 1282 y Fs(\025)2029 1226 y Fv(q)p 2026 1263 46 4 v 2026 1339 a(k)2100 1282 y Fs(\001)g Fv(T)2191 1294 y Ft(k)2231 1282 y FD(\()p Fv(S)5 b FD(\))p Black 935 w(\(32\))p Black 648 1584 a(Equation)27 b(\()1040 1584 y SDict begin H.S end 1040 1584 a Black FD(32)p Black 1123 1530 a SDict begin H.R end 1123 1530 a 1123 1584 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.32) cvn H.B /ANN pdfmark end 1123 1584 a FD(\))i(follo)n(ws)d(since)i Fs(j)p Fv(S)1737 1596 y Ft(>T)1841 1584 y Fs(j)d(\024)g Fv(\034)k Fs(\000)19 b FD(1)g(+)g Fv(q)28 b Fs(\024)d(d)p Fv(k)s(=)p FD(2)o Fs(e)19 b(\000)g FD(1)g(+)g Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)25 b(\024)g Fv(k)32 b FD(and)523 1683 y Fv(T)572 1695 y Ft(k)612 1683 y FD(\()p Fv(S)5 b FD(\))p Fv(=k)40 b FD(\(the)d(a)n(v)n(erage)d(v)-5 b(alue)36 b(of)h(the)g Fv(k)j FD(largest)34 b(items\))h(is)h(smaller)d(or)j(equal)f(to)i(the) 523 1783 y(a)n(v)n(erage)25 b(v)-5 b(alue)26 b(of)i(the)g(items)e (greater)f(than)j Fv(T)12 b FD(,)27 b(\()2113 1721 y Fu(P)2201 1808 y Ft(x)p Fq(2)p Ft(S)2325 1816 y Fo(>T)2433 1783 y Fv(v)s FD(\()p Fv(x)p FD(\))p Fv(=)p Fs(j)p Fv(S)2703 1795 y Ft(>T)2808 1783 y Fs(j)p FD(\).)648 1921 y(Since)c Fv(q)j Fs(\024)d(d)p Fv(k)s(=)p FD(2)o Fs(e)o FD(,)i Fv(C)1321 1933 y Ft(k)q(;\015)1444 1921 y FD(c)n(ho)r(oses)e(a)h Fv(\015)5 b FD(-indep)r(enden)n(t)23 b(subset)i(of)f(size)f(at)h(least) f(as)g(large)523 2021 y(as)32 b(the)i(size)d(of)i(the)h(maxim)n(um)29 b Fv(\015)5 b FD(-indep)r(enden)n(t)32 b(subset)h(of)g(those)f(among)f (the)j Fv(q)i FD(items)523 2120 y(arriving)24 b(in)j(the)h(in)n(terv)-5 b(al)25 b([1)p Fv(=)p FD(2)17 b(+)h Fv(\015)5 b(;)14 b FD(1\).)648 2251 y(Fixing)22 b(the)i(arriv)-5 b(al)21 b(times)i(of)h(the)h Fv(q)i FD(items)c(in)g Fv(S)2183 2220 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2409 2251 y FD(whic)n(h)h(are)f(ab)r(o)n(v)n(e)g(the)i(thresh-)523 2350 y(old,)19 b(the)h(assignmen)n(t)d(of)i(items)f(to)i(these)g(times) e(is)g(a)i(random)e(p)r(erm)n(utation.)f(This)i(implies)523 2450 y(that)26 b(the)f(exp)r(ected)h(v)-5 b(alue)24 b(of)h(the)h(items) d(in)i(the)g Fv(\015)5 b FD(-indep)r(enden)n(t)25 b(subset)g(pic)n(k)n (ed)f(b)n(y)h Fv(C)3306 2462 y Ft(\015)t(;k)523 2549 y FD(equals)30 b(to)i(the)g(cardinalit)n(y)c(of)k(this)f Fv(\015)5 b FD(-indep)r(enden)n(t)31 b(subset)h(divided)e(b)n(y)i Fv(q)s FD(,)g(m)n(ultiplied)523 2649 y(b)n(y)25 b(the)h(exp)r(ected)g (v)-5 b(alue)24 b(of)i(these)f(items.)f(Lemma)f(\()2205 2649 y SDict begin H.S end 2205 2649 a Black FD(9)p Black 2247 2596 a SDict begin H.R end 2247 2596 a 2247 2649 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.9) cvn H.B /ANN pdfmark end 2247 2649 a FD(\))j(giv)n(es)d(a)j(lo)n(w)n(er)d(b)r (ound)i(on)h(the)g(ex-)523 2749 y(p)r(ected)i(size)d(of)i(the)h Fv(\015)5 b FD(-indep)r(enden)n(t)26 b(subset)h(pic)n(k)n(ed)e(b)n(y)i Fv(C)2409 2761 y Ft(k)q(;\015)2536 2749 y FD(and)f(Equation)f(\()3086 2749 y SDict begin H.S end 3086 2749 a Black FD(32)p Black 3170 2695 a SDict begin H.R end 3170 2695 a 3170 2749 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.32) cvn H.B /ANN pdfmark end 3170 2749 a FD(\))i(giv)n(es)523 2848 y(a)g(lo)n(w)n(er)e(b)r(ound)j(on)g(the)g(exp)r(ected)f(v)-5 b(alue)27 b(of)g(the)h Fv(q)j FD(items,)26 b(therefore:)804 3112 y SDict begin H.S end 804 3112 a 804 3112 a SDict begin 12 H.A end 804 3112 a 804 3112 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.33) cvn /DEST pdfmark end 804 3112 a FD(E)861 3020 y Fu(h)900 3112 y Fv(C)965 3069 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))959 3137 y Ft(k)q(;\015)1167 3112 y FD(\()p Fv(S)5 b FD(\))23 b Fs(j)g Fv(Q)g FD(=)g Fv(q)1573 3020 y Fu(i)827 3320 y Fs(\025)925 3263 y FD(1)p 925 3301 42 4 v 926 3377 a Fv(q)995 3320 y Fs(\001)1036 3203 y Fu(\024)1206 3263 y Fv(q)p 1090 3301 273 4 v 1090 3377 a FD(1)18 b(+)g(2)p Fv(\015)5 b(q)1372 3227 y Fu(\020)1422 3320 y FD(1)18 b Fs(\000)1565 3247 y(p)p 1634 3247 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2033 3227 y Fu(\021)2081 3203 y(\025)2144 3320 y Fs(\001)2185 3227 y Fu(h)2237 3263 y Fv(q)p 2234 3301 46 4 v 2234 3377 a(k)2309 3320 y Fs(\001)g Fv(T)2399 3332 y Ft(k)2439 3320 y FD(\()p Fv(S)5 b FD(\))2559 3227 y Fu(i)p Black 3257 3320 a FD(\(33\))p Black 827 3541 a Fs(\025)1168 3485 y FD(1)p 925 3522 527 4 v 925 3598 a(1)18 b(+)g Fv(\015)5 b FD(\()p Fv(k)21 b FD(+)d(1)p Fv(=)p FD(2\))1462 3449 y Fu(\020)1512 3541 y FD(1)g Fs(\000)1655 3469 y(p)p 1724 3469 42 4 v 72 x FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2122 3449 y Fu(\021)2189 3541 y Fs(\001)2231 3449 y Fu(h)2283 3485 y Fv(q)p 2280 3522 46 4 v 2280 3598 a(k)2354 3541 y Fs(\001)h Fv(T)2445 3553 y Ft(k)2485 3541 y FD(\()p Fv(S)5 b FD(\))2605 3449 y Fu(i)827 3766 y FD(=)1023 3710 y(1)p 925 3747 237 4 v 925 3823 a(1)18 b(+)g Fv(\015)5 b(k)1190 3766 y Fs(\001)1387 3710 y FD(1)18 b(+)g Fv(\015)5 b(k)p 1241 3747 527 4 v 1241 3823 a FD(1)18 b(+)g Fv(\015)5 b FD(\()p Fv(k)22 b FD(+)c(1)p Fv(=)p FD(2\))1778 3674 y Fu(\020)1828 3766 y FD(1)g Fs(\000)1971 3693 y(p)p 2040 3693 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2439 3674 y Fu(\021)n(h)2539 3710 y Fv(q)p 2536 3747 46 4 v 2536 3823 a(k)2611 3766 y Fs(\001)g Fv(T)2701 3778 y Ft(k)2742 3766 y FD(\()p Fv(S)5 b FD(\))2862 3674 y Fu(i)p Black 3257 3766 a FD(\(34\))p Black 827 3987 a(=)1023 3931 y(1)p 925 3968 237 4 v 925 4044 a(1)18 b(+)g Fv(\015)5 b(k)1190 3987 y Fs(\001)1596 3931 y FD(1)p 1241 3968 750 4 v 1241 4044 a(1)18 b(+)g Fv(\015)5 b(=)p FD(2)17 b Fs(\001)i FD(1)p Fv(=)p FD(\(1)e(+)h Fv(\015)5 b(k)s FD(\)\))2001 3895 y Fu(\020)2051 3987 y FD(1)18 b Fs(\000)2194 3914 y(p)p 2263 3914 42 4 v 73 x FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2661 3895 y Fu(\021)o(h)2762 3931 y Fv(q)p 2759 3968 46 4 v 2759 4044 a(k)2834 3987 y Fs(\001)g Fv(T)2924 3999 y Ft(k)2964 3987 y FD(\()p Fv(S)5 b FD(\))3084 3895 y Fu(i)827 4207 y Fs(\025)1023 4151 y FD(1)p 925 4188 237 4 v 925 4264 a(1)18 b(+)g Fv(\015)5 b(k)1190 4207 y Fs(\001)1357 4151 y FD(1)p 1241 4188 274 4 v 1241 4264 a(1)18 b(+)g Fv(\015)5 b(=)p FD(2)1525 4115 y Fu(\020)1574 4207 y FD(1)18 b Fs(\000)1717 4135 y(p)p 1787 4135 42 4 v 1787 4207 a FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2185 4115 y Fu(\021)o(h)2286 4151 y Fv(q)p 2283 4188 46 4 v 2283 4264 a(k)2357 4207 y Fs(\001)h Fv(T)2448 4219 y Ft(k)2488 4207 y FD(\()p Fv(S)5 b FD(\))2608 4115 y Fu(i)827 4428 y Fs(\025)1023 4372 y FD(1)p 925 4409 237 4 v 925 4485 a(1)18 b(+)g Fv(\015)5 b(k)1190 4428 y Fs(\001)18 b FD(\(1)h Fs(\000)f Fv(\015)5 b(=)p FD(2\))1571 4336 y Fu(\020)1619 4428 y FD(1)18 b Fs(\000)1762 4355 y(p)p 1831 4355 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2230 4336 y Fu(\021)o(h)2331 4372 y Fv(q)p 2328 4409 46 4 v 2328 4485 a(k)2402 4428 y Fs(\001)g Fv(T)2492 4440 y Ft(k)2533 4428 y FD(\()p Fv(S)5 b FD(\))2653 4336 y Fu(i)p Black 3257 4428 a FD(\(35\))p Black 827 4655 a Fs(\025)1023 4598 y FD(1)p 925 4635 237 4 v 925 4711 a(1)18 b(+)g Fv(\015)5 b(k)1171 4562 y Fu(\020)1221 4655 y FD(1)18 b Fs(\000)1364 4582 y(p)p 1433 4582 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)1897 4562 y Fu(\021)1945 4537 y(\022)2006 4655 y FD(1)g Fs(\000)g FD(2)g Fs(\001)2261 4598 y(j)p Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)o Fs(ej)p 2261 4635 391 4 v 2433 4711 a Fv(k)2661 4537 y Fu(\023)2732 4598 y Fv(T)2781 4610 y Ft(k)2821 4598 y FD(\()p Fv(S)5 b FD(\))p 2732 4635 210 4 v 2816 4711 a(2)827 4887 y Fs(\025)1023 4831 y FD(1)p 925 4868 237 4 v 925 4944 a(1)18 b(+)g Fv(\015)5 b(k)1171 4795 y Fu(\020)1221 4887 y FD(1)18 b Fs(\000)1364 4814 y(p)p 1433 4814 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)1897 4795 y Fu(\021)1945 4770 y(\022)2006 4887 y FD(1)g Fs(\000)g FD(2)g Fs(\001)2261 4831 y(j)p Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)o Fs(ej)p 2261 4868 391 4 v 2433 4944 a Fv(k)2679 4887 y Fs(\000)2774 4831 y FD(1)p 2772 4868 46 4 v 2772 4944 a Fv(k)2828 4770 y Fu(\023)2899 4831 y Fv(T)2948 4843 y Ft(k)2988 4831 y FD(\()p Fv(S)5 b FD(\))p 2899 4868 210 4 v 2983 4944 a(2)p Black 1922 5173 a(25)p Black eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.26) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(where)37 b(Inequalit)n(y)1176 448 y SDict begin H.S end 1176 448 a Black FD(34)p Black 1259 395 a SDict begin H.R end 1259 395 a 1259 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.33) cvn H.B /ANN pdfmark end 1259 448 a 38 w FD(follo)n(w)e(since)i Fv(q)43 b Fs(\024)d(d)p Fv(k)s(=)p FD(2)o Fs(e)e FD(and)f(Inequalit)n (y)f(\()2793 448 y SDict begin H.S end 2793 448 a Black FD(35)p Black 2876 395 a SDict begin H.R end 2876 395 a 2876 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.33) cvn H.B /ANN pdfmark end 2876 448 a FD(\))i(follo)n(ws)d(since)578 515 y FC(1)p 533 529 123 4 v 533 577 a(1)p Fq(\000)p Ft(x)689 548 y Fs(\025)22 b FD(1)c(+)g Fv(x)28 b FD(for)f(an)n(y)g Fv(x)p FD(.)h(In)g(addition)e(w)n(e)h(ha)n(v)n(e)f(that)i(for)f(an)n(y) g Fv(\015)h Fs(\024)22 b Fv(\015)2791 518 y Fq(\003)995 724 y Fu(\020)1045 816 y FD(1)c Fs(\000)1188 743 y(p)p 1257 743 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)1721 724 y Fu(\021)1792 816 y Fs(\025)1880 724 y Fu(\020)1929 816 y FD(1)g Fs(\000)2072 743 y(p)p 2141 743 V 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)2316 781 y Fq(\003)2354 816 y FD(\))h Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2643 781 y Fq(\003)2680 724 y Fu(\021)2753 816 y Fv(>)k FD(0)27 b Fv(;)523 1064 y FD(therefore)35 b(all)f(terms)h(in)h(the)h(previous)d (calculations)e(are)j(p)r(ositiv)n(e)f(meaning)g(that)i(de-)523 1164 y(creasing)25 b(an)n(y)i(of)h(them)f(decreases)g(the)h(whole)e (pro)r(duct.)i(This)f(concludes)f(the)i(pro)r(of)g(for)523 1263 y Fv(q)e Fs(\024)d(d)p Fv(k)s(=)p FD(2)o Fs(e)o FD(.)648 1386 y(F)-7 b(or)32 b Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)g Fv(<)g(q)j Fs(\024)d FD(2)p Fs(d)p Fv(k)s(=)p FD(2)n Fs(e)p FD(,)i(the)f(exp)r(ected)h(sum)e(of)h(the)h(v)-5 b(alues)31 b(of)j(the)f Fv(q)j FD(items)c(in)523 1485 y Fv(S)579 1455 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))799 1485 y Fs(\\)19 b Fv(S)924 1497 y Ft(>T)1056 1485 y FD(is)523 1734 y SDict begin H.S end 523 1734 a 523 1734 a SDict begin 12 H.A end 523 1734 a 523 1734 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.36) cvn /DEST pdfmark end 523 1734 a FD(E)580 1642 y Fu(h)619 1734 y Fv(v)s FD(\()p Fs(f)p Fv(x)k Fs(j)g Fv(x)h Fs(2)f Fv(S)1057 1700 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))1277 1734 y Fs(\\)c Fv(S)1402 1746 y Ft(>T)1506 1734 y Fs(g)p FD(\))k Fs(j)g Fv(Q)g FD(=)g Fv(q)1866 1642 y Fu(i)1928 1734 y Fs(\025)2063 1656 y Fu(X)2016 1834 y Ft(x)p Fq(2)p Ft(S)2140 1842 y Fo(>T)2244 1734 y FD(Prob)2420 1642 y Fu(h)2473 1734 y Fv(x)h Fs(2)f Fv(S)2678 1700 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))2903 1734 y Fs(j)g Fv(Q)g FD(=)f Fv(q)3179 1642 y Fu(i)3237 1734 y Fs(\001)d Fv(v)s FD(\()p Fv(x)p FD(\))1928 1980 y(=)2106 1923 y Fv(q)p 2026 1960 202 4 v 2026 2037 a Fs(j)p Fv(S)2100 2049 y Ft(>T)2204 2037 y Fs(j)2298 1901 y Fu(X)2251 2079 y Ft(x)p Fq(2)p Ft(S)2375 2087 y Fo(>T)2479 1980 y Fv(v)s FD(\()p Fv(x)p FD(\))1928 2225 y Fs(\025)2171 2169 y Fv(q)p 2026 2206 331 4 v 2026 2282 a(q)i FD(+)d Fv(\034)28 b Fs(\000)19 b FD(1)2427 2146 y Fu(X)2380 2324 y Ft(x)p Fq(2)p Ft(S)2504 2332 y Fo(>T)2608 2225 y Fv(v)s FD(\()p Fv(x)p FD(\))1928 2470 y Fs(\025)2171 2414 y Fv(q)p 2026 2451 V 2026 2527 a(q)i FD(+)d Fv(\034)28 b Fs(\000)19 b FD(1)2366 2470 y Fv(T)2415 2482 y Ft(k)2455 2470 y FD(\()p Fv(S)5 b FD(\))28 b Fv(:)p Black 660 w FD(\(36\))p Black 523 2753 a(If)608 2683 y Fu(\014)608 2733 y(\014)636 2753 y Fv(S)692 2723 y FC([0)p Ft(;)p FC(1)p Ft(=)p FC(2\))893 2683 y Fu(\014)893 2733 y(\014)947 2753 y Fv(<)e(\034)39 b FD(the)30 b(threshold)e(is)h(zero)f(so)h Fv(S)2043 2765 y Ft(>T)2173 2753 y FD(=)d Fv(S)34 b FD(and)c(therefore)e(Inequalit)n(y)f(\()3289 2753 y SDict begin H.S end 3289 2753 a Black FD(36)p Black 3373 2700 a SDict begin H.R end 3373 2700 a 3373 2753 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.36) cvn H.B /ANN pdfmark end 3373 2753 a FD(\))523 2853 y(follo)n(ws.)20 b(Otherwise)h Fs(j)p Fv(S)1276 2865 y Ft(>T)1380 2853 y Fs(j)i FD(=)g Fv(\034)18 b Fs(\000)9 b FD(1)g(+)g Fv(q)25 b Fs(\025)e(d)p Fv(k)s(=)p FD(2)n Fs(e)9 b(\000)g FD(1)g(+)g Fs(d)o Fv(k)s(=)p FD(2)n Fs(e)g FD(+)g(1)22 b Fs(\025)h Fv(k)j FD(so)c Fv(S)2980 2865 y Ft(>T)3107 2853 y FD(con)n(tains)523 2953 y(the)28 b Fv(k)j FD(largest)25 b(items)h(of)h Fv(S)5 b FD(,)28 b(and)f(therefore)g(Inequalit)n(y)e(\()2365 2953 y SDict begin H.S end 2365 2953 a Black FD(36)p Black 2448 2899 a SDict begin H.R end 2448 2899 a 2448 2953 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.36) cvn H.B /ANN pdfmark end 2448 2953 a FD(\))j(follo)n(ws.)648 3075 y(Let)k Fv(s)g FD(b)r(e)g(the)g(size)f(of)h(the)g(maxim)n(um)c Fv(\015)5 b FD(-indep)r(enden)n(t)31 b(subset)h(of)g(the)g(the)h Fv(q)i FD(items)523 3175 y(larger)25 b(than)i Fv(T)39 b FD(arriving)24 b(in)j(the)h(in)n(terv)-5 b(al)25 b([1)p Fv(=)p FD(2)17 b(+)h Fv(\015)5 b(;)14 b FD(1\).)648 3297 y(Since)38 b Fv(C)935 3309 y Ft(k)q(;\015)1074 3297 y FD(is)g(restricted)g(to)h(c)n(ho)r(ose)g(at)g(most)f Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)h FD(items)f(in)h(the)h(time)e(in)n(ter-) 523 3397 y(v)-5 b(al)35 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))35 b(the)h(size)f(of)i(the)f Fv(\015)5 b FD(-indep)r(enden)n (t)36 b(subset)g(that)h Fv(C)2596 3409 y Ft(k)q(;\015)2731 3397 y FD(c)n(ho)r(oses)e(is)g(at)h(least)523 3496 y(min)n(\()p Fv(s;)14 b Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(\).)28 b(Therefore,)742 3749 y SDict begin H.S end 742 3749 a 742 3749 a SDict begin 12 H.A end 742 3749 a 742 3749 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.37) cvn /DEST pdfmark end 742 3749 a FD(E)799 3657 y Fu(h)837 3654 y(\014)837 3704 y(\014)837 3753 y(\014)865 3749 y Fv(C)930 3706 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))924 3774 y Ft(k)q(;\015)1132 3749 y FD(\()p Fv(S)5 b FD(\))1252 3654 y Fu(\014)1252 3704 y(\014)1252 3753 y(\014)1303 3749 y Fs(j)23 b Fv(Q)g FD(=)g Fv(q)1566 3657 y Fu(i)1628 3749 y Fs(\025)g FD(E[min)n(\()p Fv(s;)14 b Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(\))23 b Fs(j)g Fv(Q)g FD(=)f Fv(q)s FD(])1628 3957 y(=)h(E)1773 3840 y Fu(\024)1924 3901 y Fv(s)c Fs(\001)f(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1826 3938 498 4 v 1826 4014 a FD(max)o(\()p Fv(s;)c Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)o FD(\))2357 3957 y Fs(j)23 b Fv(Q)g FD(=)f Fv(q)2619 3840 y Fu(\025)1628 4187 y Fs(\025)h FD(E[)p Fv(s)f Fs(j)i Fv(Q)e FD(=)h Fv(q)s FD(])18 b Fs(\001)2213 4131 y(d)p Fv(k)s(=)p FD(2)o Fs(e)p 2213 4168 203 4 v 2294 4244 a Fv(q)1628 4414 y Fs(\025)1716 4297 y Fu(\022)1903 4358 y Fv(q)p 1787 4395 273 4 v 1787 4471 a FD(1)g(+)g(2)p Fv(\015)5 b(q)2069 4322 y Fu(\020)2119 4414 y FD(1)18 b Fs(\000)2262 4341 y(p)p 2331 4341 42 4 v 73 x FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2729 4322 y Fu(\021)2778 4297 y(\023)2858 4414 y Fs(\001)2909 4358 y(d)p Fv(k)s(=)p FD(2)o Fs(e)p 2909 4395 203 4 v 2990 4471 a Fv(q)1628 4646 y Fs(\025)1716 4529 y Fu(\022)1903 4590 y Fv(q)p 1787 4627 273 4 v 1787 4703 a FD(1)g(+)g(2)p Fv(\015)5 b(q)2069 4554 y Fu(\020)2119 4646 y FD(1)18 b Fs(\000)2262 4574 y(p)p 2331 4574 42 4 v 72 x FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2794 4554 y Fu(\021)2843 4529 y(\023)2922 4646 y Fs(\001)2974 4590 y(d)p Fv(k)s(=)p FD(2)n Fs(e)p 2974 4627 203 4 v 3055 4703 a Fv(q)p Black 3257 4646 a FD(\(37\))p Black 523 4924 a(where)27 b(Equation)f(\()1154 4924 y SDict begin H.S end 1154 4924 a Black FD(37)p Black 1237 4871 a SDict begin H.R end 1237 4871 a 1237 4924 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.37) cvn H.B /ANN pdfmark end 1237 4924 a FD(\))i(follo)n(ws)d(from)h(Lemma)2062 4924 y SDict begin H.S end 2062 4924 a Black FD(9)p Black 2104 4871 a SDict begin H.R end 2104 4871 a 2104 4924 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.9) cvn H.B /ANN pdfmark end 2104 4924 a FD(.)p Black 1922 5173 a(26)p Black eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.27) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Com)n(bining)26 b(Equations)i(\()1498 448 y SDict begin H.S end 1498 448 a Black FD(36)p Black 1581 395 a SDict begin H.R end 1581 395 a 1581 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.36) cvn H.B /ANN pdfmark end 1581 448 a FD(\))j(and)f(\()1840 448 y SDict begin H.S end 1840 448 a Black FD(37)p Black 1923 395 a SDict begin H.R end 1923 395 a 1923 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.37) cvn H.B /ANN pdfmark end 1923 448 a FD(\))h(using)e(the)h(same)f(argumen)n(t)f (that)i(w)n(e)g(used)523 548 y(to)e(deriv)n(e)d(Equation)h(\()1260 548 y SDict begin H.S end 1260 548 a Black FD(33)p Black 1343 494 a SDict begin H.R end 1343 494 a 1343 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.33) cvn H.B /ANN pdfmark end 1343 548 a FD(\))i(w)n(e)f(get)704 755 y SDict begin H.S end 704 755 a 704 755 a SDict begin 12 H.A end 704 755 a 704 755 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.38) cvn /DEST pdfmark end 704 755 a FD(E)761 663 y Fu(h)799 755 y Fv(C)864 712 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))858 780 y Ft(k)q(;\015)1066 755 y FD(\()p Fv(S)5 b FD(\))24 b Fs(j)f Fv(Q)g FD(=)f Fv(q)1472 663 y Fu(i)750 963 y Fs(\025)848 907 y FD(1)p 848 944 42 4 v 849 1020 a Fv(q)918 963 y Fs(\001)959 846 y Fu(\024\022)1190 907 y Fv(q)p 1074 944 273 4 v 1074 1020 a FD(1)c(+)g(2)p Fv(\015)5 b(q)1356 871 y Fu(\020)1406 963 y FD(1)18 b Fs(\000)1549 890 y(p)p 1618 890 42 4 v 73 x FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2081 871 y Fu(\021)2130 846 y(\023\025)2253 963 y Fs(\001)2295 846 y Fu(\024)2348 907 y Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p 2348 944 203 4 v 2430 1020 a Fv(q)2579 963 y Fs(\001)2776 907 y Fv(q)p 2631 944 331 4 v 2631 1020 a(q)j FD(+)d Fv(\034)28 b Fs(\000)19 b FD(1)2971 963 y Fv(T)3020 975 y Ft(k)3060 963 y FD(\()p Fv(S)5 b FD(\))3180 846 y Fu(\025)p Black 3257 963 a FD(\(38\))p Black 648 1195 a(A)28 b(simple)c(arithmetic)h(manipulation)e(\(for)k (more)f(details)g(see)h(Lemma)2966 1195 y SDict begin H.S end 2966 1195 a Black FD(10)p Black 3049 1141 a SDict begin H.R end 3049 1141 a 3049 1195 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.10) cvn H.B /ANN pdfmark end 3049 1195 a FD(\))h(giv)n(es)e(us)1392 1452 y SDict begin H.S end 1392 1452 a 1392 1452 a SDict begin 12 H.A end 1392 1452 a 1392 1452 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.39) cvn /DEST pdfmark end 1392 1452 a 1466 1396 a Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1402 1433 331 4 v 1402 1509 a Fv(\034)i FD(+)18 b Fv(q)k Fs(\000)c FD(1)1765 1452 y Fs(\025)1863 1396 y FD(1)p 1863 1433 42 4 v 1863 1509 a(2)1914 1310 y Fu( )1980 1452 y FD(1)g Fs(\000)2133 1391 y Fv(q)j Fs(\000)2274 1323 y Fu(\006)2324 1358 y Ft(k)p 2324 1372 37 4 v 2326 1419 a FC(2)2370 1323 y Fu(\007)p 2133 1433 277 4 v 2248 1509 a Fv(k)2419 1310 y Fu(!)2513 1452 y Fv(:)721 b FD(\(39\))523 1709 y(Substituting)27 b(\()1031 1709 y SDict begin H.S end 1031 1709 a Black FD(39)p Black 1114 1656 a SDict begin H.R end 1114 1656 a 1114 1709 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.39) cvn H.B /ANN pdfmark end 1114 1709 a FD(\))h(in)n(to)e(\()1374 1709 y SDict begin H.S end 1374 1709 a Black FD(38)p Black 1457 1656 a SDict begin H.R end 1457 1656 a 1457 1709 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.38) cvn H.B /ANN pdfmark end 1457 1709 a FD(\))i(w)n(e)g(obtain)523 1966 y SDict begin H.S end 523 1966 a 523 1966 a SDict begin 12 H.A end 523 1966 a 523 1966 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.40) cvn /DEST pdfmark end 523 1966 a FD(E)580 1874 y Fu(h)619 1966 y Fv(C)684 1923 y FC([1)p Ft(=)p FC(2)p Ft(;)p FC(1\))678 1991 y Ft(k)q(;\015)886 1966 y FD(\()p Fv(S)5 b FD(\))23 b Fs(j)g Fv(Q)g FD(=)g Fv(q)1292 1874 y Fu(i)1354 1966 y Fs(\025)1567 1910 y FD(1)p 1452 1947 273 4 v 1452 2023 a(1)18 b(+)g(2)p Fv(\015)5 b(q)1734 1874 y Fu(\020)1783 1966 y FD(1)18 b Fs(\000)1926 1894 y(p)p 1995 1894 42 4 v 72 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(:)p FD(5)p Fv(\015)2459 1874 y Fu(\021)2524 1966 y Fs(\001)2573 1910 y FD(1)p 2573 1947 V 2573 2023 a(2)2624 1824 y Fu( )2690 1966 y FD(1)g Fs(\000)2843 1905 y Fv(q)k Fs(\000)2985 1838 y Fu(\006)3034 1872 y Ft(k)p 3034 1886 37 4 v 3036 1933 a FC(2)3080 1838 y Fu(\007)p 2843 1947 277 4 v 2958 2023 a Fv(k)3130 1824 y Fu(!)3195 1966 y Fv(T)3244 1978 y Ft(k)3285 1966 y FD(\()p Fv(S)5 b FD(\))3257 2141 y(\(40\))523 2240 y(Another)28 b(simple)e(arithmetic)f(manipulation)g(\(for)j(more)f (details)f(see)i(Lemma)3085 2240 y SDict begin H.S end 3085 2240 a Black FD(11)p Black 3168 2187 a SDict begin H.R end 3168 2187 a 3168 2240 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.11) cvn H.B /ANN pdfmark end 3168 2240 a FD(\))h(giv)n(es)523 2340 y(us)822 2551 y SDict begin H.S end 822 2551 a 822 2551 a SDict begin 12 H.A end 822 2551 a 822 2551 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.41) cvn /DEST pdfmark end 822 2551 a 947 2495 a FD(1)p 832 2532 273 4 v 832 2608 a(1)18 b(+)g(2)p Fv(\015)5 b(q)1114 2434 y Fu(\022)1175 2551 y FD(1)18 b Fs(\000)1328 2495 y Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1328 2532 345 4 v 1477 2608 a Fv(k)1682 2434 y Fu(\023)1767 2551 y Fs(\025)1962 2495 y FD(1)p 1865 2532 237 4 v 1865 2608 a(1)g(+)g Fv(\015)5 b(k)2111 2434 y Fu(\022)2172 2551 y FD(1)18 b Fs(\000)g FD(2)g Fs(\001)2426 2495 y Fv(q)k Fs(\000)c Fv(k)s(=)p FD(2)p 2426 2532 271 4 v 2539 2608 a Fv(k)2707 2434 y Fu(\023)p Black 3257 2551 a FD(\(41\))p Black 1767 2784 a Fs(\025)1962 2728 y FD(1)p 1865 2765 237 4 v 1865 2841 a(1)g(+)g Fv(\015)5 b(k)2111 2667 y Fu(\022)2172 2784 y FD(1)18 b Fs(\000)g FD(2)g Fs(\001)2426 2728 y(j)p Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(ej)p 2426 2765 391 4 v 2599 2841 a Fv(k)2845 2784 y Fs(\000)2940 2728 y FD(1)p 2938 2765 46 4 v 2938 2841 a Fv(k)2994 2667 y Fu(\023)3083 2784 y Fv(:)p Black 151 w FD(\(42\))p Black 523 3016 a(By)27 b(substituting)f(\()1147 3016 y SDict begin H.S end 1147 3016 a Black FD(42)p Black 1231 2962 a SDict begin H.R end 1231 2962 a 1231 3016 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.41) cvn H.B /ANN pdfmark end 1231 3016 a FD(\))i(in)n(to)e(\()1491 3016 y SDict begin H.S end 1491 3016 a Black FD(40)p Black 1574 2962 a SDict begin H.R end 1574 2962 a 1574 3016 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.40) cvn H.B /ANN pdfmark end 1574 3016 a FD(\))i(the)g(lemma)d(follo)n(ws.)1034 b Fs(u)-55 b(t)523 3111 y SDict begin H.S end 523 3111 a 523 3111 a SDict begin 12 H.A end 523 3111 a 523 3111 a SDict begin [/View [/XYZ H.V]/Dest (lemma.7) cvn /DEST pdfmark end 523 3111 a Black 79 x Fr(Lemma)30 b(7.)p Black 40 w Fp(F)-6 b(or)30 b(any)g(set)g Fv(S)k Fp(and)d(any)f Fv(q)c Fs(\024)c(j)p Fv(S)5 b Fs(j)19 b(\000)f Fv(\034)1430 3372 y FD(Prob)o([)c Fv(Q)23 b FD(=)f Fv(q)17 b FD(])23 b(=)g(Prob)o([)14 b Fv(G)23 b FD(=)g Fv(q)16 b FD(])p Fv(:)523 3555 y Fp(\()p Fv(Q)29 b Fp(and)i Fv(G)f Fp(ar)l(e)g(de\014ne) l(d)g(b)l(efor)l(e)1568 3555 y SDict begin H.S end 1568 3555 a Black Fp(5)p Black 1611 3501 a SDict begin H.R end 1611 3501 a 1611 3555 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.5) cvn H.B /ANN pdfmark end 1611 3555 a Fp(\).)p Black 523 3729 a(Pr)l(o)l(of.)p Black 43 w FD(W)-7 b(e)30 b(rein)n(terpret)d(the)i(random)f(sc)n(hedule)g(as)g(though)h(w) n(e)g(sc)n(hedule)f(the)h(items)f(in)523 3828 y Fv(S)k FD(in)27 b(decreasing)e(order.)648 3928 y(Rep)r(eatedly)c(toss)h(a)g (fair)f(coin)g(un)n(til)g Fv(\034)32 b FD(\\tails")20 b(app)r(ear.)h(The)i(length)e(of)i(this)f(sequence)523 4028 y(is)29 b(the)i(sum)e(of)h Fv(\034)40 b FD(geometric)27 b(v)-5 b(ariables)27 b Fv(G)1899 4040 y Ft(i)1957 4028 y FD(\(n)n(um)n(b)r(er)i(of)h(consecutiv)n(e)e(\\heads"\))h(plus)g Fv(\034)523 4127 y FD(\(n)n(um)n(b)r(er)24 b(of)g(\\tails"\),)e(let)h Fv(G)h FD(=)1554 4065 y Fu(P)1642 4086 y Ft(\034)1642 4152 y(i)p FC(=1)1768 4127 y Fv(G)1833 4139 y Ft(i)1885 4127 y FD(b)r(e)h(a)f(random)f(v)-5 b(ariable)21 b(for)j(the)h(total)e (n)n(um)n(b)r(er)523 4227 y(of)28 b(\\heads")e(in)h(this)f(sequence.) 648 4327 y(T)-7 b(ra)n(v)n(erse)23 b(this)j(sequence)g(un)n(til)f(its)h (end)h(or)e Fv(S)31 b FD(is)26 b(exhausted,)g(sc)n(hedule)f(the)i(top)f (item)523 4426 y(remaining)k(in)k Fv(S)39 b FD(within)32 b(the)j(in)n(terv)-5 b(al)31 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))33 b(if)h(\\heads",)e(and)i(within)f(the)i(in)n(terv)-5 b(al)523 4526 y([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))26 b(if)h(\\tails".)648 4625 y(If)37 b Fv(S)k FD(has)36 b(not)h(b)r(een)g(exhausted,)g(place)e(remaining)e(items)i(to)i(app)r (ear)e(at)i(random)523 4725 y(times)26 b(in)h(time)f(in)n(terv)-5 b(al)25 b([)14 b(0)p Fv(;)g FD(1\).)648 4825 y(If)31 b Fv(S)36 b FD(w)n(as)30 b(exhausted)h(strictly)e(b)r(efore)i(the)g (end)h(of)f(the)g(sequence)g(|)g(w)n(e)g(ha)n(v)n(e)f(that)523 4924 y Fv(G)20 b FD(+)f Fv(\034)36 b(>)26 b Fs(j)p Fv(S)5 b Fs(j)29 b FD(\(total)f(n)n(um)n(b)r(er)h(of)g(coin)f(tosses\),)h(and) g(the)h(n)n(um)n(b)r(er)e(of)h(items)f(arriving)e(in)p Black 1922 5173 a(27)p Black eop end end %%Page: 28 28 TeXDict begin HPSdict begin 28 27 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.28) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(the)30 b(time)d(in)n(terv)-5 b(al)27 b([0)p Fv(;)14 b FD(1)p Fv(=)p FD(2\))28 b(is)g Fv(<)d(\034)9 b FD(,)30 b(th)n(us)g(all)d(items)g(arriving)f(during)i ([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))27 b(m)n(ust)i(b)r(e)523 548 y(coun)n(ted)e(in)g Fv(Q)h FD(and)f(therefore)g Fv(Q)22 b(>)h Fs(j)p Fv(S)5 b Fs(j)18 b(\000)g Fv(\034)38 b FD(as)27 b(w)n(ell.)648 648 y(If)19 b Fv(S)24 b FD(w)n(as)18 b(not)h(exhausted)f (strictly)f(b)r(efore)i(the)g(last)f(coin)g(toss,)g(then)i Fv(G)q FD(+)q Fv(\034)33 b Fs(\024)22 b(j)q Fv(S)5 b Fs(j)o FD(,)19 b(ergo,)523 747 y(an)n(y)24 b(item)g(placed)g(in)g(the)h (time)f(in)n(terv)-5 b(al)22 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))24 b(b)r(efore)g(the)i(last)e(coin)f(toss)i(con)n(tributes)523 847 y(to)f Fv(Q)p FD(,)g(and)h(an)n(y)e(other)h(item)f(m)n(ust)g(not)i (b)r(e)f(coun)n(ted)g(in)g Fv(Q)g FD(\(either)g(it)f(w)n(as)h(placed)f (in)g(time)523 946 y(in)n(terv)-5 b(al)17 b([0)p Fv(;)d FD(1)p Fv(=)p FD(2\))k(or)g(its)h(v)-5 b(alue)18 b(is)g(less)g(than)h (the)h(threshold)e(v)-5 b(alue)18 b(|)i(the)f(minim)n(um)c(of)20 b(the)523 1046 y Fv(\034)29 b FD(largest)16 b(items)i(sc)n(heduled)g (during)f(the)j(time)d(in)n(terv)-5 b(al)17 b([0)p Fv(;)d FD(1)p Fv(=)p FD(2\).)j(Th)n(us,)i Fv(Q)k FD(=)f Fv(G)i Fs(\024)e(j)p Fv(S)5 b Fs(j)q(\000)q Fv(\034)k FD(.)648 1146 y(W)-7 b(e)26 b(conclude)g(that)h Fv(G)c(>)g Fs(j)p Fv(S)5 b Fs(j)16 b(\000)g Fv(\034)36 b FD(i\013)26 b Fv(Q)d(>)f Fs(j)p Fv(S)5 b Fs(j)16 b(\000)g Fv(\034)37 b FD(and)26 b(for)g(an)n(y)g Fv(q)g Fs(\024)d(j)p Fv(S)5 b Fs(j)16 b(\000)g Fv(\034)9 b FD(,)27 b Fv(G)c FD(=)g Fv(q)523 1245 y FD(i\013)k Fv(Q)c FD(=)g Fv(q)s FD(.)k(So)1222 1345 y(Prob)n([)14 b Fv(G)23 b(>)g Fs(j)p Fv(S)5 b Fs(j)18 b(\000)g Fv(\034)24 b FD(])f(=)g(Prob)n([)14 b Fv(Q)23 b(>)g Fs(j)p Fv(S)5 b Fs(j)18 b(\000)g Fv(\034)23 b FD(])p Fv(;)523 1491 y FD(and)k(for)h(an)n(y)e Fv(q)g Fs(\024)d(j)p Fv(S)5 b Fs(j)18 b(\000)g Fv(\034)1430 1668 y FD(Prob)o([)c Fv(G)23 b FD(=)g Fv(q)17 b FD(])23 b(=)g(Prob)n([)14 b Fv(Q)23 b FD(=)g Fv(q)16 b FD(])p Fv(:)3350 1844 y Fs(u)-55 b(t)523 1913 y SDict begin H.S end 523 1913 a 523 1913 a SDict begin 12 H.A end 523 1913 a 523 1913 a SDict begin [/View [/XYZ H.V]/Dest (lemma.8) cvn /DEST pdfmark end 523 1913 a Black 100 x Fr(Lemma)30 b(8.)p Black 40 w Fp(F)-6 b(or)35 b(any)g(r)l(andom)g(variable)h Fv(W)12 b Fp(,)35 b(with)g(me)l(an)f Fv(\026)g Fp(and)h(standar)l(d)g (deviation)523 2112 y Fv(\033)s Fp(,)1677 2212 y FD(E)1734 2145 y Fu(\002)1768 2212 y Fs(j)p Fv(W)30 b Fs(\000)18 b Fv(\026)p Fs(j)2055 2145 y Fu(\003)2113 2212 y Fs(\024)23 b Fv(\033)p Black 523 2381 a Fp(Pr)l(o)l(of.)p Black 43 w FD(By)k(the)h(de\014nition)e(of)i(the)g(v)-5 b(ariance)1164 2557 y(V)e(ar)1294 2490 y Fu(\002)1328 2557 y Fs(j)p Fv(W)31 b Fs(\000)18 b Fv(\026)p Fs(j)1616 2490 y Fu(\003)1674 2557 y FD(=)k(E)1818 2490 y Fu(\002)1852 2557 y Fs(j)p Fv(W)31 b Fs(\000)18 b Fv(\026)p Fs(j)2140 2523 y FC(2)2177 2490 y Fu(\003)2230 2557 y Fs(\000)g FD(E)2370 2523 y FC(2)2407 2490 y Fu(\002)2442 2557 y Fs(j)p Fv(W)30 b Fs(\000)18 b Fv(\026)p Fs(j)2729 2490 y Fu(\003)1674 2691 y FD(=)k(V)-7 b(ar)1891 2624 y Fu(\002)1925 2691 y Fv(W)2015 2624 y Fu(\003)2068 2691 y Fs(\000)18 b FD(E)2208 2657 y FC(2)2245 2624 y Fu(\002)2280 2691 y Fs(j)p Fv(W)30 b Fs(\000)18 b Fv(\026)p Fs(j)2567 2624 y Fu(\003)523 2876 y FD(Since)27 b(V)-7 b(ar)869 2809 y Fu(\002)904 2876 y Fs(j)p Fv(W)30 b Fs(\000)18 b Fv(\026)p Fs(j)1191 2809 y Fu(\003)1249 2876 y Fs(\025)23 b FD(0)k(it)g(follo)n(ws)d(that) 1415 3063 y(E)1472 3028 y FC(2)1508 2995 y Fu(\002)1543 3063 y Fs(j)p Fv(W)30 b Fs(\000)18 b Fv(\026)p Fs(j)1830 2995 y Fu(\003)1888 3063 y Fs(\024)23 b FD(V)-7 b(ar)2105 2995 y Fu(\002)2140 3063 y Fv(W)2230 2995 y Fu(\003)2287 3063 y FD(=)23 b Fv(\033)2425 3028 y FC(2)2490 3063 y Fv(:)523 3239 y FD(The)28 b(lemma)c(follo)n(ws)h(b)n(y)i(taking)f(the)i (square)e(ro)r(ot)h(of)h(b)r(oth)g(sides.)676 b Fs(u)-55 b(t)523 3324 y SDict begin H.S end 523 3324 a 523 3324 a SDict begin 12 H.A end 523 3324 a 523 3324 a SDict begin [/View [/XYZ H.V]/Dest (lemma.9) cvn /DEST pdfmark end 523 3324 a Black 84 x Fr(Lemma)30 b(9.)p Black 40 w Fp(L)l(et)25 b FD(0)d Fv(<)h(\015)28 b Fs(\024)22 b FD(1)j Fp(and)g(let)g Fv(\011)34 b Fp(b)l(e)25 b(a)h(set)e(of)i(items)f(sche)l (dule)l(d)h(at)f(uniform)h(times)523 3507 y(during)33 b(the)f(interval)h FD([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\))p Fp(,)31 b(and)i(let)f Fv( )e FD(=)e Fs(j)p Fv(\011)9 b Fs(j)p Fp(.)32 b(L)l(et)g Fv(\011)2380 3477 y Fq(0)2436 3507 y Fp(b)l(e)g(the)g(subset)g(of)h Fv(\011)41 b Fp(that)33 b(ar)l(e)523 3607 y(sche)l(dule)l(d)26 b(during)f(the)f(time)h (interval)g FD([1)p Fv(=)p FD(2)7 b(+)g Fv(\015)e(;)14 b FD(1\))p Fp(.)24 b(The)h(exp)l(e)l(cte)l(d)g(size)g(of)g(the)g (maximum)523 3707 y Fv(\015)5 b Fp(-indep)l(endent)30 b(subset)f(of)h(items)g(fr)l(om)g Fv(\011)1883 3676 y Fq(0)1936 3707 y Fp(is)h(at)e(le)l(ast)1569 3866 y Fv( )p 1453 3903 290 4 v 1453 3979 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1752 3830 y Fu(\020)1802 3922 y FD(1)18 b Fs(\000)1945 3849 y(p)p 2014 3849 42 4 v 73 x FD(2)p Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2413 3830 y Fu(\021)2461 3922 y Fv(;)523 4158 y Fp(wher)l(e)30 b Fv(\013)p FD(\()p Fv(\015)5 b FD(\))24 b(=)f(3)1076 4087 y Fu(p)p 1158 4087 326 4 v 1158 4158 a Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))29 b Fp(\(as)h(de\014ne)l(d)g(in)g (The)l(or)l(em)2381 4158 y SDict begin H.S end 2381 4158 a Black Fp(3)p Black 2423 4105 a SDict begin H.R end 2423 4105 a 2423 4158 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2423 4158 a Fp(\).)p Black 523 4327 a(Pr)l(o)l(of.)p Black 43 w FD(If)24 b Fv(\015)k Fs(\025)23 b FD(1)p Fv(=)p FD(2)f(the)i(lemma)c(holds)j (since)f(the)i(b)r(ound)g(is)e(negativ)n(e,)g(therefore)h(assume)523 4426 y Fv(\015)28 b(<)22 b FD(1)p Fv(=)p FD(2.)648 4526 y(Let)30 b Fv(\032)f FD(b)r(e)i(the)f(size)e(of)i(the)g(maxim)n(um)c Fv(\015)5 b FD(-indep)r(enden)n(t)29 b(subset)h(of)g Fv(\011)9 b FD(,)30 b(and)f(let)g Fv(\032)3266 4496 y Fq(0)3320 4526 y FD(b)r(e)523 4625 y(the)h(size)f(of)g(the)h(maximal)25 b Fv(\015)5 b FD(-indep)r(enden)n(t)29 b(subset)h(of)g Fv(\011)2372 4595 y Fq(0)2425 4625 y FD(\(b)r(oth)g Fv(\032)g FD(and)f Fv(\032)2934 4595 y Fq(0)2987 4625 y FD(are)g(random)523 4725 y(v)-5 b(ariables\).)25 b(It)j(is)e(easy)h(to)g(see)g(that)h Fv(\032)1737 4695 y Fq(0)1784 4725 y Fs(\025)22 b Fv(\032)d Fs(\000)f FD(1.)648 4825 y(Let)33 b Fv(I)40 b FD(b)r(e)33 b(an)g(indicator)d(v)-5 b(ariable)30 b(where)j Fv(I)39 b FD(=)32 b(1)g(if)h(some)e(item)h(in)g Fv(\011)42 b FD(is)32 b(sc)n(heduled)523 4924 y(during)26 b(the)i(time)e(in)n(terv) -5 b(al)25 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1)p Fv(=)p FD(2)i(+)i Fv(\015)5 b FD(\),)27 b(and)h Fv(I)i FD(=)23 b(0)k(otherwise.)p Black 1922 5173 a(28)p Black eop end end %%Page: 29 29 TeXDict begin HPSdict begin 29 28 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.29) cvn /DEST pdfmark end 515 125 a Black Black 711 729 a SDict begin H.S end 711 729 a 711 729 a SDict begin 12 H.A end 711 729 a 711 729 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.43) cvn /DEST pdfmark end 711 729 a FD(E[)p Fv(\032)834 694 y Fq(0)857 729 y FD(])23 b Fs(\025)g FD(Prob)o([)14 b Fv(I)30 b FD(=)22 b(0)14 b(])k Fs(\001)h FD(E[)o Fv(\032)k Fs(j)g Fv(I)31 b FD(=)22 b(0])74 b(+)f(Prob)o([)13 b Fv(I)31 b FD(=)22 b(1)14 b(])p Fs(\001)2588 661 y Fu(\000)2626 729 y FD(E[)o Fv(\032)23 b Fs(j)g Fv(I)30 b FD(=)23 b(1])18 b Fs(\000)g FD(1)3179 661 y Fu(\001)903 853 y FD(=)23 b(E[)p Fv(\032)p FD(])18 b Fs(\000)g FD(Prob)o([)c Fv(I)30 b FD(=)22 b(1)14 b(])p Fv(:)523 1133 y FD(T)-7 b(o)37 b(b)r(ound)i(E[)o Fv(\032)p FD(])f(w)n(e)f(recall)e(that)j Fv(\011)47 b FD(consists)36 b(of)i Fv( )j FD(p)r(oin)n(ts)36 b(randomly)f(placed)i(in)g(the)523 1233 y(time)e(in)n(terv)-5 b(al)34 b([1)p Fv(=)p FD(2)p Fv(;)14 b FD(1\).)35 b(Ev)n(ery)g Fv(\015)5 b FD(-indep)r(enden)n(t)36 b(subset)g(of)h Fv(\011)46 b FD(is)35 b(analogous)e(to)k(a)f(2)p Fv(\015)5 b FD(-)523 1343 y(indep)r(enden)n(t)22 b(subset)g(of)1338 1322 y Fu(e)1326 1343 y Fv(\011)32 b FD(where)1655 1322 y Fu(e)1643 1343 y Fv(\011)g FD(=)23 b Fs(f)p FD(2)p Fv(p)8 b Fs(\000)g FD(1)20 b Fs(j)j Fv(p)g Fs(2)g Fv(\011)9 b Fs(g)p FD(.)22 b Fw(I.e.)p FD(,)g(the)h(size)e(of)h(the)h(maximal)523 1443 y Fv(\015)5 b FD(-indep)r(enden)n(t)32 b(subset)g(of)h Fv(\011)41 b FD(is)32 b(equal)f(to)h(the)h(size)e(of)i(the)f(maximal)c (2)p Fv(\015)5 b FD(-indep)r(enden)n(t)523 1542 y(subset)28 b(of)884 1521 y Fu(e)872 1542 y Fv(\011)9 b FD(.)28 b(Ergo,)e(w)n(e)h (can)g(apply)f(Theorem)2059 1542 y SDict begin H.S end 2059 1542 a Black FD(3)p Black 2101 1489 a SDict begin H.R end 2101 1489 a 2101 1542 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2101 1542 a 27 w FD(and)i(hence:)523 1861 y SDict begin H.S end 523 1861 a 523 1861 a SDict begin 12 H.A end 523 1861 a 523 1861 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.43) cvn /DEST pdfmark end 523 1861 a FD(E[)p Fv(\032)p FD(])18 b Fs(\000)g FD(Prob)o([)c Fv(I)30 b FD(=)22 b(1)14 b(])23 b Fs(\025)1453 1805 y Fv( )p 1337 1842 290 4 v 1337 1918 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1636 1861 y FD(\(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\)\))19 b Fs(\000)f FD(Prob)o([)c Fv(I)30 b FD(=)22 b(1)14 b(])1239 2081 y(=)1453 2025 y Fv( )p 1337 2062 V 1337 2138 a FD(1)k(+)g(2)p Fv(\015)5 b( )1636 1989 y Fu(\020)1685 2081 y FD(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\))2035 1989 y Fu(\021)2104 2081 y Fs(\000)18 b FD(\(1)g Fs(\000)g FD(\(1)g Fs(\000)g FD(2)p Fv(\015)5 b FD(\))2659 2047 y Ft( )2709 2081 y FD(\))1239 2314 y(=)1453 2258 y Fv( )p 1337 2295 V 1337 2371 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1636 2197 y Fu(\022)1697 2314 y FD(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\))19 b Fs(\000)2158 2258 y FD(\(1)g Fs(\000)f FD(\(1)g Fs(\000)g FD(2)p Fv(\015)5 b FD(\))2631 2228 y Ft( )2681 2258 y FD(\)\(1)18 b(+)g(2)p Fv(\015)5 b( )s FD(\))p 2158 2295 909 4 v 2584 2371 a Fv( )3077 2197 y Fu(\023)1239 2553 y FD(=)1453 2497 y Fv( )p 1337 2534 290 4 v 1337 2610 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1636 2436 y Fu(\022)1697 2553 y FD(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\))19 b Fs(\000)2158 2497 y FD(\(1)g Fs(\000)f FD(\(1)g Fs(\000)g FD(2)p Fv(\015)5 b FD(\))2631 2467 y Ft( )2681 2497 y FD(\))p 2158 2534 555 4 v 2407 2610 a Fv( )2741 2553 y Fs(\000)18 b FD(2)p Fv(\015)5 b FD(\(1)18 b Fs(\000)g FD(\(1)g Fs(\000)g FD(2)p Fv(\015)5 b FD(\))3386 2519 y Ft( )3436 2553 y FD(\))3468 2436 y Fu(\023)1239 2786 y Fs(\025)1453 2729 y Fv( )p 1337 2767 290 4 v 1337 2843 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1636 2669 y Fu(\022)1697 2786 y FD(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\))19 b Fs(\000)2158 2729 y FD(1)f Fs(\000)g FD(\(1)h Fs(\000)f FD(2)p Fv(\015)5 b( )s FD(\))p 2158 2767 497 4 v 2378 2843 a Fv( )2684 2786 y Fs(\000)18 b FD(2)p Fv(\015)2857 2669 y Fu(\023)p Black 3381 2786 a FD(\(43\))p Black 1239 3012 a(=)1453 2955 y Fv( )p 1337 2992 290 4 v 1337 3068 a FD(1)g(+)g(2)p Fv(\015)5 b( )1636 3012 y FD(\(1)18 b Fs(\000)g Fv(\013)p FD(\(2)p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)5 b FD(\))1239 3232 y(=)1453 3175 y Fv( )p 1337 3213 V 1337 3289 a FD(1)18 b(+)g(2)p Fv(\015)5 b( )1636 3139 y Fu(\020)1685 3232 y FD(1)18 b Fs(\000)g FD(3)1870 3157 y Fu(p)p 1953 3157 474 4 v 75 x FD(2)p Fv(\015)g FD(ln)o(\(1)p Fv(=)p FD(\(2)p Fv(\015)5 b FD(\)\))18 b Fs(\000)g FD(4)p Fv(\015)2618 3139 y Fu(\021)1239 3452 y Fs(\025)1453 3396 y Fv( )p 1337 3433 290 4 v 1337 3509 a FD(1)g(+)g(2)p Fv(\015)5 b( )1636 3360 y Fu(\020)1685 3452 y FD(1)18 b Fs(\000)g FD(3)1870 3377 y Fu(p)p 1953 3377 368 4 v 75 x FD(2)p Fv(\015)g FD(ln)o(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g FD(4)p Fv(\015)2512 3360 y Fu(\021)1239 3672 y FD(=)1453 3616 y Fv( )p 1337 3653 290 4 v 1337 3729 a FD(1)g(+)g(2)p Fv(\015)5 b( )1636 3580 y Fu(\020)1685 3672 y FD(1)18 b Fs(\000)1828 3599 y(p)p 1898 3599 42 4 v 1898 3672 a FD(2)o Fv(\013)p FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(4)p Fv(\015)2296 3580 y Fu(\021)2345 3672 y Fv(:)523 3996 y FD(Where)41 b(the)g(Inequalit)n(y)d(\()1395 3996 y SDict begin H.S end 1395 3996 a Black FD(43)p Black 1478 3943 a SDict begin H.R end 1478 3943 a 1478 3996 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.43) cvn H.B /ANN pdfmark end 1478 3996 a FD(\))j(follo)n(ws)d(since)h(for)h(an)n(y)g Fv(n)45 b Fs(2)g FD(N)c(and)f(an)n(y)g Fv(x)45 b Fs(2)g FD(\(0)p Fv(;)14 b FD(1\),)523 4096 y(\(1)k Fs(\000)g Fv(x)p FD(\))777 4066 y Ft(n)846 4096 y Fv(>)23 b FD(\(1)18 b Fs(\000)g Fv(nx)p FD(\).)2089 b Fs(u)-55 b(t)523 4338 y SDict begin H.S end 523 4338 a 523 4338 a SDict begin 12 H.A end 523 4338 a 523 4338 a SDict begin [/View [/XYZ H.V]/Dest (lemma.10) cvn /DEST pdfmark end 523 4338 a Black 128 x Fr(Lemma)30 b(10.)p Black 40 w Fp(F)-6 b(or)30 b(any)g Fv(k)c Fs(2)d FD(N)30 b Fp(and)h(any)f Fv(q)c Fs(\025)d(d)o Fv(k)s(=)p FD(2)o Fs(e)1432 4764 y(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1369 4801 331 4 v 1369 4878 a Fv(\034)28 b FD(+)18 b Fv(q)j Fs(\000)d FD(1)1732 4821 y Fs(\025)1830 4764 y FD(1)p 1830 4801 42 4 v 1830 4878 a(2)1881 4679 y Fu( )1947 4821 y FD(1)g Fs(\000)2100 4759 y FD(\()p Fv(q)j Fs(\000)2273 4692 y Fu(\006)2323 4726 y Ft(k)p 2323 4740 37 4 v 2325 4788 a FC(2)2369 4692 y Fu(\007)2408 4759 y FD(\))p 2100 4801 342 4 v 2247 4878 a Fv(k)2451 4679 y Fu(!)2546 4821 y Fv(:)p Black 1922 5173 a FD(29)p Black eop end end %%Page: 30 30 TeXDict begin HPSdict begin 30 29 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.30) cvn /DEST pdfmark end 515 125 a Black Black Black 523 448 a Fp(Pr)l(o)l(of.)p Black 1267 676 a SDict begin H.S end 1267 676 a 1267 676 a SDict begin 12 H.A end 1267 676 a 1267 676 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.44) cvn /DEST pdfmark end 1267 676 a 1340 620 a Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1277 657 331 4 v 1277 733 a Fv(\034)28 b FD(+)18 b Fv(q)j Fs(\000)d FD(1)1640 676 y(=)1880 620 y Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1738 657 488 4 v 1738 733 a(d)p Fv(k)s(=)p FD(2)o Fs(e)g FD(+)g Fv(q)j Fs(\000)d FD(1)1640 905 y(=)2085 849 y Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1738 886 898 4 v 1738 962 a FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)g FD(+)g(\()p Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(\))h Fs(\000)f FD(1)1640 1135 y Fs(\025)2014 1079 y(d)p Fv(k)s(=)p FD(2)n Fs(e)p 1738 1116 755 4 v 1738 1192 a FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)g FD(+)g(\()p Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(\))1640 1391 y(=)1728 1249 y Fu( )1804 1335 y FD(1)p 1804 1372 42 4 v 1804 1448 a(2)1873 1391 y Fs(\000)2089 1297 y FC(1)p 2089 1311 34 4 v 2089 1358 a(2)2132 1330 y FD(\()p Fv(q)k Fs(\000)2306 1262 y Fu(\006)2355 1297 y Ft(k)p 2355 1311 37 4 v 2357 1358 a FC(2)2402 1262 y Fu(\007)2441 1330 y FD(\))p 1966 1372 620 4 v 1966 1455 a(2)2008 1388 y Fu(\006)2057 1422 y Ft(k)p 2057 1436 37 4 v 2059 1484 a FC(2)2104 1388 y Fu(\007)2161 1455 y FD(+)c(\()p Fv(q)k Fs(\000)2418 1388 y Fu(\006)2467 1422 y Ft(k)p 2467 1436 V 2469 1484 a FC(2)2514 1388 y Fu(\007)2553 1455 y FD(\))2596 1249 y Fu(!)1640 1673 y Fs(\025)1728 1531 y Fu( )1804 1617 y FD(1)p 1804 1654 42 4 v 1804 1730 a(2)1873 1673 y Fs(\000)1966 1617 y FD(1)p 1966 1654 V 1966 1730 a(2)2036 1673 y Fs(\001)2088 1612 y Fv(q)f Fs(\000)2229 1545 y Fu(\006)2279 1579 y Ft(k)p 2279 1593 37 4 v 2281 1640 a FC(2)2325 1545 y Fu(\007)p 2088 1654 277 4 v 2138 1737 a FD(2)2180 1670 y Fu(\006)2229 1704 y Ft(k)p 2229 1718 37 4 v 2231 1766 a FC(2)2275 1670 y Fu(\007)2374 1531 y(!)p Black 3257 1673 a FD(\(44\))p Black 1640 1956 a Fs(\025)1738 1899 y FD(1)p 1738 1936 42 4 v 1738 2012 a(2)1789 1814 y Fu( )1855 1956 y FD(1)d Fs(\000)2008 1894 y Fv(q)j Fs(\000)2149 1827 y Fu(\006)2199 1861 y Ft(k)p 2199 1875 37 4 v 2201 1923 a FC(2)2245 1827 y Fu(\007)p 2008 1936 277 4 v 2029 2012 a FD(2\()p Fv(k)s(=)p FD(2\))2294 1814 y Fu(!)p Black 3257 1956 a FD(\(45\))p Black 1640 2238 a Fs(\025)1738 2182 y FD(1)p 1738 2219 42 4 v 1738 2295 a(2)1789 2096 y Fu( )1855 2238 y FD(1)d Fs(\000)2008 2176 y Fv(q)j Fs(\000)2149 2109 y Fu(\006)2199 2144 y Ft(k)p 2199 2158 37 4 v 2201 2205 a FC(2)2245 2109 y Fu(\007)p 2008 2219 277 4 v 2123 2295 a Fv(k)2294 2096 y Fu(!)2388 2238 y Fv(:)p Black 846 w FD(\(46\))p Black 523 2497 a(where)27 b(\()795 2497 y SDict begin H.S end 795 2497 a Black FD(44)p Black 878 2444 a SDict begin H.R end 878 2444 a 878 2497 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.44) cvn H.B /ANN pdfmark end 878 2497 a FD(\),)h(\()993 2497 y SDict begin H.S end 993 2497 a Black FD(45)p Black 1077 2444 a SDict begin H.R end 1077 2444 a 1077 2497 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.44) cvn H.B /ANN pdfmark end 1077 2497 a FD(\))g(and)f(\()1330 2497 y SDict begin H.S end 1330 2497 a Black FD(46)p Black 1413 2444 a SDict begin H.R end 1413 2444 a 1413 2497 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.44) cvn H.B /ANN pdfmark end 1413 2497 a FD(\))h(follo)n(w)d(since)i Fv(q)f Fs(\025)c(d)p Fv(k)s(=)p FD(2)o Fs(e)p FD(.)1057 b Fs(u)-55 b(t)648 2675 y FD(The)36 b(follo)n(wing)d(Lemma)h(is)i(used) g(in)g(the)h(pro)r(of)f(of)h(Lemma)2676 2675 y SDict begin H.S end 2676 2675 a Black FD(6)p Black 2718 2622 a SDict begin H.R end 2718 2622 a 2718 2675 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.6) cvn H.B /ANN pdfmark end 2718 2675 a FD(,)g(for)f(b)r(ounding)g(the)523 2775 y(pro\014t)h(of)g(the)h(algorithm)33 b(from)i(items)h(in)g(the)i(2nd)f (half,)f(and)h(conditioned)e(on)i(there)523 2874 y(b)r(eing)27 b Fv(q)j FD(suc)n(h)e(items)e(ab)r(o)n(v)n(e)g(the)i(threshold.)523 2890 y SDict begin H.S end 523 2890 a 523 2890 a SDict begin 12 H.A end 523 2890 a 523 2890 a SDict begin [/View [/XYZ H.V]/Dest (lemma.11) cvn /DEST pdfmark end 523 2890 a Black 163 x Fr(Lemma)i(11.)p Black 40 w Fp(F)-6 b(or)41 b(any)h Fv(k)k Fs(2)e FD(N)p Fp(,)e Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)h(\024)g Fv(q)k Fs(\024)c FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)e Fp(and)h(any)f Fv(\015)46 b Fp(that)41 b(satis\014es)523 3153 y Fv(k)26 b Fs(\024)d FD(1)p Fv(=\015)33 b Fp(we)d(have)980 3387 y SDict begin H.S end 980 3387 a 980 3387 a SDict begin 12 H.A end 980 3387 a 980 3387 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.47) cvn /DEST pdfmark end 980 3387 a 1105 3331 a FD(1)p 990 3368 273 4 v 990 3444 a(1)18 b(+)g(2)p Fv(\015)5 b(q)1272 3270 y Fu(\022)1333 3387 y FD(1)18 b Fs(\000)1486 3331 y Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1486 3368 345 4 v 1635 3444 a Fv(k)1840 3270 y Fu(\023)1924 3387 y Fs(\025)2119 3331 y FD(1)p 2022 3368 237 4 v 2022 3444 a(1)g(+)g Fv(\015)5 b(k)2268 3270 y Fu(\022)2329 3387 y FD(1)18 b Fs(\000)g FD(2)g Fs(\001)2584 3331 y Fv(q)j Fs(\000)d Fv(k)s(=)p FD(2)p 2584 3368 271 4 v 2696 3444 a Fv(k)2864 3270 y Fu(\023)2925 3387 y Fv(:)309 b FD(\(47\))p Black 523 3623 a Fp(Pr)l(o)l(of.)p Black 43 w FD(While)28 b(the)i(factor)f(on)g(the)h(left)g(in)f(the)h(left)f(hand)g(side)g(of)g (\()2719 3623 y SDict begin H.S end 2719 3623 a Black FD(47)p Black 2803 3569 a SDict begin H.R end 2803 3569 a 2803 3623 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2803 3623 a FD(\))h(is)e(smaller)e(than) 523 3722 y(factor)21 b(on)h(the)g(left)g(in)f(the)i(righ)n(t)d(hand)i (side)f(of)h(\()2050 3722 y SDict begin H.S end 2050 3722 a Black FD(47)p Black 2133 3669 a SDict begin H.R end 2133 3669 a 2133 3722 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2133 3722 a FD(\),)h(w)n(e)f(sho)n(w)f(that)h(this)f(is)g(comp)r(ensated)523 3822 y(for)27 b(the)h(factors)f(on)g(the)h(righ)n(t)e(of)h(b)r(oth)h (sides)f(of)g(the)h(Inequalit)n(y)-7 b(.)648 3922 y(Note)27 b(that)h(for)f(an)n(y)g Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)c(\024)f Fv(q)k Fs(\024)d FD(2)p Fs(d)o Fv(k)s(=)p FD(2)o Fs(e)p FD(,)1394 4155 y SDict begin H.S end 1394 4155 a 1394 4155 a SDict begin 12 H.A end 1394 4155 a 1394 4155 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.48) cvn /DEST pdfmark end 1394 4155 a FD(1)18 b Fs(\000)1547 4099 y Fv(q)k Fs(\000)c(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1547 4136 345 4 v 1697 4212 a Fv(k)1925 4155 y Fs(\025)23 b FD(1)18 b Fs(\000)2166 4099 y(d)o Fv(k)s(=)p FD(2)o Fs(e)p 2166 4136 203 4 v 2244 4212 a Fv(k)p Black 3257 4155 a FD(\(48\))p Black 1925 4363 a Fs(\025)23 b FD(1)18 b Fs(\000)2166 4307 y Fv(k)s(=)p FD(2)f(+)h(1)p Fv(=)p FD(2)p 2166 4344 355 4 v 2320 4420 a Fv(k)1925 4516 y FD(=)23 b(1)p Fv(=)p FD(2)17 b Fs(\001)h FD(\(1)g Fs(\000)g FD(1)p Fv(=k)s FD(\))1925 4640 y Fs(\025)23 b FD(0)p Fv(:)p Black 1179 w FD(\(49\))p Black 523 4825 a(The)i(deriv)-5 b(ation)23 b(\()1111 4825 y SDict begin H.S end 1111 4825 a Black FD(48)p Black 1194 4771 a SDict begin H.R end 1194 4771 a 1194 4825 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.48) cvn H.B /ANN pdfmark end 1194 4825 a FD(\))i(ab)r(o)n(v)n(e)f (follo)n(ws)e(b)n(y)j(substituting)e(the)j(maximal)20 b(v)-5 b(alue)24 b Fv(q)k FD(can)d(tak)n(e.)523 4924 y(In)30 b(Equation)d(\()1021 4924 y SDict begin H.S end 1021 4924 a Black FD(47)p Black 1104 4871 a SDict begin H.R end 1104 4871 a 1104 4924 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 1104 4924 a FD(\))j(b)r(oth)g(factors)e(1)p Fv(=)p FD(\(1)19 b(+)g(2)p Fv(\015)5 b(q)s FD(\))29 b(\(on)g(the)h(left\))f(and)g(1)p Fv(=)p FD(\(1)19 b(+)g Fv(\015)5 b(k)s FD(\))29 b(\(on)h(the)p Black 1922 5173 a(30)p Black eop end end %%Page: 31 31 TeXDict begin HPSdict begin 31 30 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.31) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(righ)n(t\))30 b(are)h(strictly)e(p)r(ositiv) n(e)g(since)i(b)r(oth)h Fv(q)j FD(and)c Fv(k)k FD(are)30 b(strictly)g(p)r(ositiv)n(e.)f(It)j(therefore)523 548 y(follo)n(ws)21 b(from)h(\()1016 548 y SDict begin H.S end 1016 548 a Black FD(49)p Black 1100 494 a SDict begin H.R end 1100 494 a 1100 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.48) cvn H.B /ANN pdfmark end 1100 548 a FD(\))i(that)g(the)g(left)f(hand)h(side)f(of)h(Equation)d(\()2456 548 y SDict begin H.S end 2456 548 a Black FD(47)p Black 2540 494 a SDict begin H.R end 2540 494 a 2540 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2540 548 a FD(\))j(is)f(non)g(negativ)n(e.)f(It)i (also)523 648 y(follo)n(ws)f(that)k(if)f(1)15 b Fs(\000)h FD(2\()p Fv(q)j Fs(\000)c Fv(k)s(=)p FD(2\))p Fv(=k)25 b Fs(\024)e FD(0)j(then)g(the)h(righ)n(t)e(hand)h(side)f(of)i(Equation) d(\()3207 648 y SDict begin H.S end 3207 648 a Black FD(47)p Black 3290 594 a SDict begin H.R end 3290 594 a 3290 648 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 3290 648 a FD(\))j(is)523 747 y Fs(\024)c FD(0)k(and)g(the)h(lemma)d(holds.)648 847 y(Th)n(us,)32 b(w)n(e)h(ma)n(y)e(assume)g(that)i(the)g(righ)n(t)f(hand) g(side)g(of)h(Equation)e(\()2911 847 y SDict begin H.S end 2911 847 a Black FD(47)p Black 2994 793 a SDict begin H.R end 2994 793 a 2994 847 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2994 847 a FD(\))i(is)f(strictly)523 946 y(p)r(ositiv)n(e,)25 b(and)j(th)n(us)f(that)1609 1089 y SDict begin H.S end 1609 1089 a 1609 1089 a SDict begin 12 H.A end 1609 1089 a 1609 1089 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.50) cvn /DEST pdfmark end 1609 1089 a FD(1)18 b Fs(\000)g FD(2)g Fs(\001)1863 1033 y Fv(q)k Fs(\000)c Fv(k)s(=)p FD(2)p 1863 1070 271 4 v 1976 1146 a Fv(k)2167 1089 y(>)k FD(0)p Fv(:)938 b FD(\(50\))523 1258 y(As)28 b Fv(q)e Fs(\025)c(d)p Fv(k)s(=)p FD(2)o Fs(e)28 b FD(it)f(follo)n(ws)d(that)k Fv(q)22 b Fs(\000)c Fv(k)s(=)p FD(2)k Fs(\025)g FD(0)28 b(and)f(th)n(us)h(inequalit)n(y)c(\()2774 1258 y SDict begin H.S end 2774 1258 a Black FD(50)p Black 2857 1204 a SDict begin H.R end 2857 1204 a 2857 1258 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.50) cvn H.B /ANN pdfmark end 2857 1258 a FD(\))k(implies)c(that)1660 1473 y SDict begin H.S end 1660 1473 a 1660 1473 a SDict begin 12 H.A end 1660 1473 a 1660 1473 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.51) cvn /DEST pdfmark end 1660 1473 a FD(1)18 b Fs(\000)1813 1417 y Fv(q)j Fs(\000)d Fv(k)s(=)p FD(2)p 1813 1454 271 4 v 1925 1530 a Fv(k)2116 1473 y(>)23 b FD(0)p Fv(:)988 b FD(\(51\))648 1668 y(As)25 b Fs(d)p Fv(k)s(=)p FD(2)o Fs(e)e(\025)g Fv(k)s(=)p FD(2)h(w)n(e)h(can) g(b)r(ound)h(the)g(left)g(hand)f(side)g(of)g(Equation)f(\()2896 1668 y SDict begin H.S end 2896 1668 a Black FD(47)p Black 2979 1615 a SDict begin H.R end 2979 1615 a 2979 1668 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2979 1668 a FD(\))i(as)f(follo)n(ws:)1013 1885 y SDict begin H.S end 1013 1885 a 1013 1885 a SDict begin 12 H.A end 1013 1885 a 1013 1885 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.52) cvn /DEST pdfmark end 1013 1885 a 1138 1829 a FD(1)p 1023 1866 273 4 v 1023 1942 a(1)18 b(+)g(2)p Fv(\015)5 b(q)1305 1768 y Fu(\022)1366 1885 y FD(1)18 b Fs(\000)1519 1829 y Fv(q)j Fs(\000)d(d)p Fv(k)s(=)p FD(2)o Fs(e)p 1519 1866 345 4 v 1668 1942 a Fv(k)1873 1768 y Fu(\023)1957 1885 y Fs(\025)2170 1829 y FD(1)p 2055 1866 273 4 v 2055 1942 a(1)g(+)g(2)p Fv(\015)5 b(q)2337 1768 y Fu(\022)2398 1885 y FD(1)18 b Fs(\000)2551 1829 y Fv(q)j Fs(\000)d Fv(k)s(=)p FD(2)p 2551 1866 271 4 v 2663 1942 a Fv(k)2831 1768 y Fu(\023)2892 1885 y Fv(:)342 b FD(\(52\))523 2103 y(As)26 b(the)g(righ)n(t)e(hand)i(side)f (of)g(Equation)f(\()1836 2103 y SDict begin H.S end 1836 2103 a Black FD(47)p Black 1919 2049 a SDict begin H.R end 1919 2049 a 1919 2103 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 1919 2103 a FD(\))i(is)f(strictly)f(p)r(ositiv)n(e,)f(w)n(e)j(can)f(m)n(ultiply)d (and)523 2202 y(divide)k(the)i(righ)n(t)e(hand)h(side)g(of)g(\()1614 2202 y SDict begin H.S end 1614 2202 a Black FD(52)p Black 1698 2149 a SDict begin H.R end 1698 2149 a 1698 2202 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.52) cvn H.B /ANN pdfmark end 1698 2202 a FD(\))h(b)n(y)f(the)h(righ)n(t)e(hand)h (side)g(of)g(Equation)f(\()3076 2202 y SDict begin H.S end 3076 2202 a Black FD(47)p Black 3159 2149 a SDict begin H.R end 3159 2149 a 3159 2202 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 3159 2202 a FD(\):)839 2420 y SDict begin H.S end 839 2420 a 839 2420 a SDict begin 12 H.A end 839 2420 a 839 2420 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.53) cvn /DEST pdfmark end 839 2420 a 965 2363 a FD(1)p 849 2401 273 4 v 849 2477 a(1)18 b(+)g(2)p Fv(\015)5 b(q)1132 2303 y Fu(\022)1193 2420 y FD(1)18 b Fs(\000)1346 2363 y Fv(q)j Fs(\000)d Fv(k)s(=)p FD(2)p 1346 2401 271 4 v 1458 2477 a Fv(k)1626 2303 y Fu(\023)1710 2420 y FD(=)1906 2363 y(1)p 1808 2401 237 4 v 1808 2477 a(1)g(+)g Fv(\015)5 b(k)2054 2303 y Fu(\022)2116 2420 y FD(1)18 b Fs(\000)g FD(2)g Fs(\001)2370 2363 y Fv(q)j Fs(\000)d Fv(k)s(=)p FD(2)p 2370 2401 271 4 v 2482 2477 a Fv(k)2650 2303 y Fu(\023)p Black 3257 2420 a FD(\(53\))p Black 1964 2677 a Fs(\001)1987 2560 y Fu(\022)2076 2621 y FD(1)g(+)g Fv(\015)5 b(k)p 2058 2658 273 4 v 2058 2734 a FD(1)18 b(+)g(2)p Fv(\015)5 b(q)2340 2560 y Fu(\023)2420 2677 y Fs(\001)2461 2535 y Fu( )2588 2616 y FD(1)18 b Fs(\000)2741 2576 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 2741 2597 189 4 v 2817 2645 a Ft(k)p 2537 2658 453 4 v 2537 2752 a FD(1)g Fs(\000)g FD(2)g Fs(\001)2792 2711 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 2792 2733 189 4 v 2867 2780 a Ft(k)3000 2535 y Fu(!)3065 2677 y Fv(:)523 2919 y FD(Note)g(the)h(the)g(righ)n(t) e(hand)h(side)f(ab)r(o)n(v)n(e)g(in)h(Equation)e(\()2224 2919 y SDict begin H.S end 2224 2919 a Black FD(53)p Black 2308 2866 a SDict begin H.R end 2308 2866 a 2308 2919 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.53) cvn H.B /ANN pdfmark end 2308 2919 a FD(\))i(is)g(of)g(the)h(form)e (RHS\(Equation)3399 2919 y SDict begin H.S end 3399 2919 a Black FD(47)p Black 3482 2866 a SDict begin H.R end 3482 2866 a 3482 2919 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 3482 2919 a FD(\))523 3019 y(times)29 b(some)g(factor)g Fv(t)p FD(.)i(By)f (assumption)e(RHS)q(\(Equation)2419 3019 y SDict begin H.S end 2419 3019 a Black FD(47)p Black 2502 2965 a SDict begin H.R end 2502 2965 a 2502 3019 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.47) cvn H.B /ANN pdfmark end 2502 3019 a FD(\))g Fv(>)f FD(0)j(so,)g(giv)n(en)f(Equation)523 3118 y(\()555 3118 y SDict begin H.S end 555 3118 a Black FD(52)p Black 638 3065 a SDict begin H.R end 638 3065 a 638 3118 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.52) cvn H.B /ANN pdfmark end 638 3118 a FD(\))f(and)g(that)g Fv(t)23 b Fs(\025)f FD(1)27 b(then)i(the)f(lemma)c(holds,)i(so)h(it)g (su\016ces)h(to)f(sho)n(w)g(that)h Fv(t)23 b Fs(\025)f FD(1:)1415 3397 y SDict begin H.S end 1415 3397 a 1415 3397 a SDict begin 12 H.A end 1415 3397 a 1415 3397 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.2.54) cvn /DEST pdfmark end 1415 3397 a 1442 3341 a FD(1)c(+)g Fv(\015)5 b(k)p 1424 3378 273 4 v 1424 3454 a FD(1)18 b(+)g(2)p Fv(\015)5 b(q)1725 3397 y Fs(\001)1827 3219 y Fu(\020)1877 3311 y FD(1)18 b Fs(\000)2030 3271 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 2030 3292 189 4 v 2105 3339 a Ft(k)2228 3219 y Fu(\021)p 1776 3378 552 4 v 1776 3391 a(\020)1826 3484 y FD(1)g Fs(\000)g FD(2)g Fs(\001)2080 3443 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 2080 3464 189 4 v 2156 3512 a Ft(k)2279 3391 y Fu(\021)2361 3397 y Fs(\025)23 b FD(1)p Fv(:)743 b FD(\(54\))523 3676 y(As)27 b Fv(\015)5 b FD(,)26 b Fv(k)s FD(,)g Fv(q)j FD(are)d(non)g(negativ)n(e,)e(and)j(giv)n(en)d (Inequalities)f(\()2400 3676 y SDict begin H.S end 2400 3676 a Black FD(50)p Black 2483 3622 a SDict begin H.R end 2483 3622 a 2483 3676 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.50) cvn H.B /ANN pdfmark end 2483 3676 a FD(\))k(and)f(\()2734 3676 y SDict begin H.S end 2734 3676 a Black FD(51)p Black 2818 3622 a SDict begin H.R end 2818 3622 a 2818 3676 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.51) cvn H.B /ANN pdfmark end 2818 3676 a FD(\),)h(it)e(follo)n(ws)f(that)523 3775 y(the)g(n)n(umerators)c(and)k (den)n(umerators,)c(in)j(b)r(oth)h(fractions)d(whose)i(pro)r(duct)g(is) f(on)i(the)f(left)523 3875 y(hand)28 b(side)e(of)i(Equation)d(\()1382 3875 y SDict begin H.S end 1382 3875 a Black FD(54)p Black 1466 3822 a SDict begin H.R end 1466 3822 a 1466 3875 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.2.54) cvn H.B /ANN pdfmark end 1466 3875 a FD(\),)j(are)e(strictly)g(p)r(ositiv)n (e.)648 3975 y(W)-7 b(e)28 b(substitute)f Fv(r)j FD(for)d Fv(q)22 b Fs(\000)c Fv(k)s(=)p FD(2,)27 b(note)g(that)h Fv(r)e Fs(\025)c FD(0,)28 b(it)f(no)n(w)g(follo)n(ws)d(that)989 4197 y(1)18 b(+)g Fv(\015)5 b(k)p 971 4234 273 4 v 971 4310 a FD(1)18 b(+)g(2)p Fv(\015)5 b(q)1271 4253 y Fs(\001)1374 4075 y Fu(\020)1423 4167 y FD(1)18 b Fs(\000)1576 4127 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 1576 4148 189 4 v 1652 4196 a Ft(k)1774 4075 y Fu(\021)p 1323 4234 552 4 v 1323 4248 a(\020)1373 4340 y FD(1)f Fs(\000)i FD(2)e Fs(\001)1627 4300 y Ft(q)r Fq(\000)p Ft(k)q(=)p FC(2)p 1627 4321 189 4 v 1703 4368 a Ft(k)1825 4248 y Fu(\021)1908 4253 y FD(=)2171 4197 y(1)h(+)g Fv(\015)5 b(k)p 2006 4234 567 4 v 2006 4310 a FD(1)18 b(+)g(2)p Fv(\015)5 b FD(\()p Fv(k)s(=)p FD(2)16 b(+)i Fv(r)r FD(\))2601 4253 y Fs(\001)2667 4125 y Fu(\000)2705 4192 y FD(1)g Fs(\000)2860 4159 y Ft(r)p 2858 4173 37 4 v 2858 4221 a(k)2904 4125 y Fu(\001)p 2652 4234 305 4 v 2652 4248 a(\000)2690 4315 y FD(1)g Fs(\000)2843 4282 y FC(2)p Ft(r)p 2843 4296 66 4 v 2858 4344 a(k)2919 4248 y Fu(\001)1908 4545 y FD(=)2121 4489 y(1)g(+)g Fv(\015)5 b(k)p 2006 4526 467 4 v 2006 4602 a FD(1)18 b(+)g Fv(\015)5 b(k)21 b FD(+)d(2)p Fv(\015)5 b(r)2501 4545 y Fs(\001)2573 4489 y FD(\()p Fv(k)22 b Fs(\000)c Fv(r)r FD(\))p 2552 4526 294 4 v 2552 4602 a(\()q Fv(k)j Fs(\000)d FD(2)p Fv(r)r FD(\))2855 4545 y Fv(:)523 4757 y FD(So)27 b(it)g(su\016ces)h(to)f(sho) n(w:)1220 4924 y(\(1)18 b(+)g Fv(\015)5 b(k)s FD(\)\()p Fv(k)22 b Fs(\000)c Fv(r)r FD(\))24 b Fs(\025)e FD(\(1)d(+)f Fv(\015)5 b(k)21 b FD(+)d(2)p Fv(\015)5 b(r)r FD(\)\()p Fv(k)21 b Fs(\000)d FD(2)p Fv(r)r FD(\))p Black 1922 5173 a(31)p Black eop end end %%Page: 32 32 TeXDict begin HPSdict begin 32 31 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.32) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(or,)27 b(equiv)-5 b(alen)n(tly)e(,)1149 632 y(\(1)18 b(+)g Fv(\015)5 b(k)s FD(\)\()p Fv(k)21 b Fs(\000)d Fv(r)r FD(\))i Fs(\000)e FD(\(1)g(+)g Fv(\015)5 b(k)21 b FD(+)d(2)p Fv(\015)5 b(r)r FD(\)\()p Fv(k)22 b Fs(\000)c FD(2)p Fv(r)r FD(\))23 b Fs(\025)g FD(0)776 898 y(\(1)18 b(+)g Fv(\015)5 b(k)s FD(\)\()p Fv(k)22 b Fs(\000)c Fv(r)r FD(\))h Fs(\000)f FD(\(1)g(+)g Fv(\015)5 b(k)21 b FD(+)d(2)p Fv(\015)5 b(r)r FD(\)\()p Fv(k)22 b Fs(\000)c FD(2)p Fv(r)r FD(\))799 1023 y(=)23 b(\(1)18 b(+)g Fv(\015)5 b(k)s FD(\))p Fv(k)21 b Fs(\000)d FD(\(1)h(+)f Fv(\015)5 b(k)s FD(\))p Fv(r)21 b Fs(\000)d FD(\(1)g(+)g Fv(\015)5 b(k)s FD(\))p Fv(k)21 b FD(+)d(2\(1)g(+)g Fv(\015)5 b(k)s FD(\))p Fv(r)21 b Fs(\000)d FD(2)p Fv(\015)5 b(r)r(k)21 b FD(+)d(4)p Fv(\015)5 b(r)3115 989 y FC(2)799 1147 y FD(=)23 b(\(1)18 b(+)g Fv(\015)5 b(k)s FD(\))p Fv(r)21 b Fs(\000)d FD(2)p Fv(\015)5 b(r)21 b Fs(\001)d Fv(k)j FD(+)d(4)p Fv(\015)5 b(r)1794 1113 y FC(2)799 1272 y FD(=)23 b Fv(r)926 1205 y Fu(\000)965 1272 y FD(1)18 b(+)g Fv(\015)5 b(k)21 b Fs(\000)d FD(2)p Fv(\015)5 b(k)20 b FD(+)e(4)p Fv(\015)5 b(r)1668 1205 y Fu(\001)799 1405 y FD(=)23 b Fv(r)926 1338 y Fu(\000)965 1405 y FD(1)18 b Fs(\000)g Fv(\015)5 b(k)21 b FD(+)d(4)p Fv(\015)5 b(r)1432 1338 y Fu(\001)799 1529 y Fs(\025)23 b FD(0)523 1713 y(where)c(the)g(last)f(inequalit)n(y)e(holds)i(since)g Fv(k)26 b Fs(\024)c FD(1)p Fv(=\015)h FD(\(and)c(therefore)g Fv(k)s(\015)27 b Fs(\024)c FD(1\))c(and)g(4)p Fv(\015)5 b(r)25 b Fs(\025)d FD(0.)3350 1812 y Fs(u)-55 b(t)523 1963 y SDict begin H.S end 523 1963 a 523 1963 a SDict begin 12 H.A end 523 1963 a 523 1963 a SDict begin [/View [/XYZ H.V]/Dest (section.A.3) cvn /DEST pdfmark end 523 1963 a 116 x Fx(C)112 b(Pro)s(of)37 b(of)h(Theorem)1628 2079 y SDict begin H.S end 1628 2079 a Black Fx(3)p Black 1684 2015 a SDict begin H.R end 1684 2015 a 1684 2079 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 1684 2079 a 523 2280 a FD(De\014ne)1011 2463 y Fv(\013)p FD(\()p Fv(x)p FD(\))25 b(=)e(3)1330 2388 y Fu(p)p 1412 2388 326 4 v 1412 2463 a Fv(x)14 b FD(ln\(1)p Fv(=x)p FD(\))p Fv(;)1147 2588 y(t)23 b FD(=)g(max)n Fs(f)p Fv(x)g Fs(j)g Fv(\013)p FD(\()p Fv(x)p FD(\))i Fs(\024)e FD(1)p Fs(g)p Fv(;)179 b(t)23 b FD(=)f(0)p Fv(:)p FD(032704)e Fs(\031)j FD(1)p Fv(=)p FD(31)p Fv(:)523 2692 y SDict begin H.S end 523 2692 a 523 2692 a SDict begin 12 H.A end 523 2692 a 523 2692 a SDict begin [/View [/XYZ H.V]/Dest (theorem.dummy.4) cvn /DEST pdfmark end 523 2692 a Black 80 x Fr(Theorem)30 b(3.)p Black 41 w Fp(L)l(et)35 b Fv(Z)j FD(=)32 b Fs(f)p Fv(z)1466 2784 y FC(1)1502 2772 y Fv(;)14 b(z)1578 2784 y FC(2)1615 2772 y Fv(;)g(:)g(:)g(:)g(;)g (z)1839 2784 y Ft(n)1883 2772 y Fs(g)35 b Fp(b)l(e)g(a)g(set)g(of)g (indep)l(endently)i(uniform)e(sam-)523 2871 y(ples,)c Fv(z)753 2883 y Ft(i)780 2871 y Fp(,)f(fr)l(om)h(the)f(r)l(e)l(al)g (interval)g FD([0)p Fv(;)14 b FD(1\))p Fp(.)30 b(F)-6 b(or)29 b FD(0)23 b Fs(\024)g Fv(\015)k Fs(\024)c FD(1)p Fp(,)641 3102 y SDict begin H.S end 641 3102 a 641 3102 a SDict begin 12 H.A end 641 3102 a 641 3102 a SDict begin [/View [/XYZ H.V]/Dest (equation.\(restate\040dummy.4\)1.4.5) cvn /DEST pdfmark end 641 3102 a FD(E)698 3114 y Ft(Z)751 3102 y FD([)p Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)1197 3046 y FD(1)18 b Fs(\000)g Fv(\013)p FD(\()p Fv(\015)5 b FD(\))p 1197 3083 309 4 v 1210 3159 a Fv(\015)24 b FD(+)18 b(1)p Fv(=n)1539 3102 y FD(=)1637 3046 y(\(1)g Fs(\000)g Fv(\013)p FD(\()p Fv(\015)5 b FD(\)\))p Fv(n)p 1637 3083 423 4 v 1728 3159 a FD(1)18 b(+)g Fv(n\015)2070 3102 y(;)98 b FD(where)27 b Fv(\013)p FD(\()p Fv(\015)5 b FD(\))24 b(=)f(3)2750 3027 y Fu(p)p 2832 3027 326 4 v 2832 3102 a Fv(\015)c FD(ln)o(\(1)p Fv(=\015)5 b FD(\))o Fv(;)118 b FD(\(5\))523 3334 y Fp(wher)l(e)30 b Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))30 b Fp(denotes)g(the)g(size)g(of) h(the)f(lar)l(gest)g Fv(\015)5 b Fp(-indep)l(endent)30 b(subset)f(of)h Fv(Z)6 b Fp(.)648 3510 y FD(W)-7 b(e)30 b(\014rst)g(giv)n(e)f(an)h(o)n(v)n(erview)d(of)k(the)f(pro)r(of)g(of)g (Theorem)2517 3510 y SDict begin H.S end 2517 3510 a Black FD(3)p Black 2559 3456 a SDict begin H.R end 2559 3456 a 2559 3510 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2559 3510 a FD(,)g(and)g(then)h(giv)n(e) e(the)h(full)523 3609 y(pro)r(of)d(in)g(detail.)523 3776 y SDict begin H.S end 523 3776 a 523 3776 a SDict begin 12 H.A end 523 3776 a 523 3776 a SDict begin [/View [/XYZ H.V]/Dest (subsection.A.3.1) cvn /DEST pdfmark end 523 3776 a 84 x Fr(C.1)95 b(Pro)s(of)31 b(Ov)m(erview)523 4027 y FD(W)-7 b(e)19 b(need)f(the)h(follo)n(wing)14 b(de\014nitions)j(for)h (the)g(pro)r(of)g(of)h(Theorem)2595 4027 y SDict begin H.S end 2595 4027 a Black FD(3)p Black 2637 3974 a SDict begin H.R end 2637 3974 a 2637 4027 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2637 4027 a FD(.)f(Let)h Fv(Z)28 b FD(=)23 b Fs(f)p Fv(z)3072 4039 y FC(1)3108 4027 y Fv(;)14 b(:)g(:)g(:)g(;)g(z)3332 4039 y Ft(n)3377 4027 y Fs(g)523 4127 y FD(and)25 b(de\014ne)h(the)g (random)e(v)-5 b(ariable)22 b Fv(X)1742 4139 y Ft(i)1770 4127 y FD(,)j(1)e Fs(\024)g Fv(i)f Fs(\024)h Fv(n)i FD(to)h(b)r(e)g (the)g Fv(i)p FD('th)g(smallest)c(p)r(oin)n(t)i(in)h Fv(Z)6 b FD(.)523 4227 y(De\014ne)29 b(the)f(random)e(v)-5 b(ariable)25 b Fv(C)1600 4239 y Ft(i)1656 4227 y FD(to)j(b)r(e)h(the)f (n)n(um)n(b)r(er)f(of)h Fv(X)2482 4239 y Ft(j)2516 4227 y FD('s)g(that)h(lie)d(in)h(the)i(in)n(terv)-5 b(al)523 4326 y([)p Fv(X)615 4338 y Ft(i)643 4326 y Fv(;)14 b(X)749 4338 y Ft(i)785 4326 y FD(+)9 b Fv(\015)c FD(\).)22 b(A)n(t)i(most)d (one)i(of)g(the)g Fv(X)1743 4338 y Ft(j)1801 4326 y Fs(2)g FD([)p Fv(X)1971 4338 y Ft(i)1999 4326 y Fv(;)14 b(X)2105 4338 y Ft(i)2141 4326 y FD(+)9 b Fv(\015)c FD(\))23 b(can)f(b)r(elong)g (to)g(a)h Fv(\015)5 b FD(-indep)r(enden)n(t)523 4426 y(set.)648 4526 y(Giv)n(en)25 b(an)n(y)h(set)h Fv(S)32 b FD(of)26 b(p)r(oin)n(ts)g(w)n(e)g(de\014ne)h(the)h(follo)n(wing)22 b Fp(gr)l(e)l(e)l(dy)30 b(algorithm)35 b FD(that)27 b(con-)523 4625 y(structs)35 b(a)f Fv(\015)5 b FD(-indep)r(enden)n(t)34 b(set.)h(The)g(greedy)f(algorithm)c(initializes)g(the)35 b(set)g(with)g(the)523 4725 y(smallest)25 b(p)r(oin)n(t)k(in)f Fv(S)33 b FD(and)c(then)g(tra)n(v)n(erses)e(the)i(remaining)c(p)r(oin)n (ts)j(in)g(increasing)d(order)523 4825 y(and)c(adds)g(the)h(p)r(oin)n (t)f Fv(x)h FD(to)f(the)h Fv(\015)5 b FD(-indep)r(enden)n(t)21 b(set)g(if)g Fv(x)h FD(is)e(larger)f(b)n(y)i(at)g(least)g Fv(\015)26 b FD(from)20 b(an)n(y)523 4924 y(p)r(oin)n(t)28 b(previously)d(added.)j(W)-7 b(e)29 b(denote)f(b)n(y)g Fv(G)p FD(\()p Fv(S;)14 b(\015)5 b FD(\))29 b(the)g Fv(\015)5 b FD(-indep)r(enden)n(t)27 b(set)i(computed)p Black 1922 5173 a(32)p Black eop end end %%Page: 33 33 TeXDict begin HPSdict begin 33 32 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.33) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(b)n(y)36 b(applying)e(greedy)h(algorithm)d (to)37 b Fv(S)5 b FD(.)36 b(Let)h Fv(I)2076 460 y Ft(i)2140 448 y FD(b)r(e)g(a)f(random)e(v)-5 b(ariable)34 b(with)i(binary)523 548 y(v)-5 b(alues)26 b(where)h Fv(I)1046 560 y Ft(i)1098 548 y FD(=)22 b(1)27 b(i\013)h Fv(X)1423 560 y Ft(i)1473 548 y Fs(2)23 b Fv(G)p FD(\()p Fv(Z)q(;)14 b(d)p FD(\),)29 b(and)f(de\014ne)f Fv(p)2313 560 y Ft(i)2364 548 y FD(=)22 b(Prob)o([)14 b Fv(I)2701 560 y Ft(i)2752 548 y FD(=)23 b(1)14 b(])o Fv(:)648 655 y FD(Lemma)956 655 y SDict begin H.S end 956 655 a Black FD(12)p Black 1039 602 a SDict begin H.R end 1039 602 a 1039 655 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.12) cvn H.B /ANN pdfmark end 1039 655 a 39 w FD(pro)n(v)n(es)37 b(the)j(w)n(ell)d(kno)n(wn)h(fact) i(that)f(the)h(greedy)e(algorithm)d(pic)n(ks)i(a)523 755 y Fv(\015)5 b FD(-indep)r(enden)n(t)34 b(set)g(of)g(largest)e (size.)h(So)h Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))35 b(=)e Fs(j)p Fv(G)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))q Fs(j)p FD(.)34 b(Our)g(pro)r(of)g(uses)g(this)523 854 y(fact.)648 962 y(W)-7 b(e)29 b(giv)n(e)d(an)j(outline)e(of)h(the)h (pro)r(of)f(of)h(Theorem)2263 962 y SDict begin H.S end 2263 962 a Black FD(3)p Black 2305 908 a SDict begin H.R end 2305 908 a 2305 962 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 2305 962 a FD(.)f(The)h(details)e(are)g(in)h(Section)3322 962 y SDict begin H.S end 3322 962 a Black FD(C)p Black 3382 905 a SDict begin H.R end 3382 905 a 3382 962 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.A.3) cvn H.B /ANN pdfmark end 3382 962 a FD(.)523 1061 y(The)g(pro)r(of)f(splits)e(in)n (to)i(t)n(w)n(o)g(main)e(sub)r(cases:)648 1168 y Fr(\\Small)g Fv(n)p Fr(":)e FD(In)h(this)f(case)g(the)h(exp)r(ected)f(n)n(um)n(b)r (er)g(of)g(non-o)n(v)n(erlapping)c(in)n(terv)-5 b(als)21 b(of)523 1268 y(the)e(form)f(\()p Fv(z)916 1280 y Ft(i)944 1268 y Fs(\000)q Fv(\015)5 b(;)14 b(z)1134 1280 y Ft(i)1160 1268 y FD(])19 b(that)g(con)n(tain)e(no)i(p)r(oin)n(t)f Fv(z)2011 1280 y Ft(j)2045 1268 y FD(,)h Fv(j)28 b Fs(6)p FD(=)23 b Fv(i)18 b FD(is)g(su\016cien)n(tly)e(large.)h(Sp)r (eci\014cally)-7 b(,)523 1378 y(Lemma)817 1378 y SDict begin H.S end 817 1378 a Black FD(13)p Black 900 1324 a SDict begin H.R end 900 1324 a 900 1378 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.13) cvn H.B /ANN pdfmark end 900 1378 a 25 w FD(sho)n(ws)24 b(that)h(for)f Fv(n)f Fs(\024)g FD(2)1665 1307 y Fu(p)p 1747 1307 453 4 v 1747 1378 a Fv(e)c Fs(\001)f FD(ln)o(\(1)p Fv(=\015)5 b FD(\))p Fv(=\015)29 b FD(the)c(exp)r(ected)g(n)n(um)n(b)r(er)f(of)h (in)n(terv)-5 b(als)523 1477 y(of)27 b(this)g(form)f(is)g(at)h(least) 1424 1445 y Ft(n)p 1363 1459 164 4 v 1363 1506 a FC(1+)p Ft(n\015)1536 1477 y FD(\(1)18 b Fs(\000)f Fv(\013)p FD(\()p Fv(\015)5 b FD(\)\).)29 b(All)c(suc)n(h)i Fv(z)2320 1489 y Ft(i)2375 1477 y FD(will)d(b)r(e)k(c)n(hosen)f(b)n(y)f(the)i (greedy)523 1577 y(algorithm)23 b(th)n(us)28 b(pro)n(ving)d(the)j (theorem)e(for)h(small)e Fv(n)p FD(.)648 1684 y Fr(\\Larger)42 b Fv(n)p Fr(":)36 b FD(F)-7 b(or)36 b Fv(n)h(>)g FD(2)1565 1613 y Fu(p)p 1648 1613 453 4 v 71 x Fv(e)18 b Fs(\001)g FD(ln\(1)p Fv(=\015)5 b FD(\))p Fv(=\015)s FD(,)36 b(w)n(e)g(need)h(to) f(consider)e(the)j(exp)r(ected)523 1784 y(n)n(um)n(b)r(er)25 b(of)h(p)r(oin)n(ts)f Fv(z)1204 1796 y Ft(j)1265 1784 y FD(\\discarded")e(when)j(the)h(greedy)e(algorithm)d(c)n(ho)r(oses)j (some)f Fv(z)3254 1796 y Ft(i)3281 1784 y FD(,)j(so)523 1883 y(the)h(k)n(ey)f(p)r(oin)n(t)g(in)g(this)f(part)i(of)f(the)h(pro)r (of)f(is)g(to)g(b)r(ound)h(E[)p Fv(C)2506 1895 y Ft(i)2557 1883 y Fs(j)23 b Fv(I)2639 1895 y Ft(i)2690 1883 y FD(=)g(1])o(.)648 1991 y(Giv)n(en)40 b(a)h(set)g(of)g(p)r(oin)n(ts)g Fv(Z)6 b FD(,)41 b(since)f(the)i(greedy)e(algorithm)d(pic)n(ks)j(a)h(maximal)c Fv(\015)5 b FD(-)523 2090 y(indep)r(enden)n(t)18 b(set)h(w)n(e)f(ha)n (v)n(e)f(that)i Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))23 b(=)1961 2028 y Fu(P)2049 2048 y Ft(n)2049 2115 y(i)p FC(=1)2174 2090 y Fv(I)2210 2102 y Ft(i)2238 2090 y FD(.)c(Therefore)e(E[)p Fv(m)p FD(\()p Fv(Z)q(;)d(\015)5 b FD(\)])23 b(=)3140 2028 y Fu(P)3228 2048 y Ft(n)3228 2115 y(i)p FC(=1)3354 2090 y Fv(p)3396 2102 y Ft(i)3423 2090 y FD(.)523 2190 y(So)k(to)h(pro)n(v)n(e)e(the)i(theorem)e(w)n(e)h (seek)g(a)g(lo)n(w)n(er)e(b)r(ound)j(for)2399 2128 y Fu(P)2487 2148 y Ft(n)2487 2215 y(i)p FC(=1)2613 2190 y Fv(p)2655 2202 y Ft(i)2682 2190 y FD(.)648 2297 y(W)-7 b(e)35 b(note)g(that)1177 2235 y Fu(P)1265 2255 y Ft(n)1265 2322 y(i)p FC(=1)1390 2297 y Fv(C)1449 2309 y Ft(i)1501 2297 y Fs(\001)23 b Fv(I)1583 2309 y Ft(i)1647 2297 y FD(=)35 b Fv(n)p FD(.)g(This)f(follo)n(ws)e(b)r(ecause)j(\(a\))g(the)g Fv(C)2997 2309 y Ft(i)3061 2297 y FD(for)f(whic)n(h)523 2397 y Fv(I)559 2409 y Ft(i)611 2397 y FD(=)24 b(1)k(are)f(non-o)n(v)n (erlapping)d(so)k(this)f(sum)h(is)f Fs(\024)d Fv(n)p FD(,)k(and)g(\(b\))2534 2334 y Fu(P)2622 2355 y Ft(n)2622 2422 y(i)p FC(=1)2747 2397 y Fv(C)2806 2409 y Ft(i)2853 2397 y Fs(\001)19 b Fv(I)2931 2409 y Ft(i)2984 2397 y Fv(<)24 b(n)k FD(implies)523 2496 y(that)k(the)h(greedy)e(algorithm)d (skipp)r(ed)j(o)n(v)n(er)g(a)h(p)r(oin)n(t)f(that)h(should)f(ha)n(v)n (e)g(b)r(een)i(c)n(hosen.)523 2596 y(In)28 b(particular,)c(the)k(exp)r (ectation)f(of)1718 2534 y Fu(P)1806 2554 y Ft(n)1806 2621 y(i)p FC(=1)1931 2596 y Fv(C)1990 2608 y Ft(i)2037 2596 y Fs(\001)18 b Fv(I)2114 2608 y Ft(i)2170 2596 y FD(is)26 b(also)g Fv(n)p FD(.)648 2703 y(Con)n(tra)n(wise,)e(it)j(m)n (ust)g(b)r(e)h(that)668 2971 y Fv(E)734 2829 y Fu( )839 2867 y Ft(n)800 2892 y Fu(X)806 3069 y Ft(i)p FC(=1)934 2971 y Fv(C)993 2983 y Ft(i)1039 2971 y Fs(\001)19 b Fv(I)1117 2983 y Ft(i)1145 2829 y Fu(!)1234 2971 y FD(=)1361 2867 y Ft(n)1321 2892 y Fu(X)1327 3069 y Ft(i)p FC(=1)1455 2971 y FD(E[)p Fv(C)1594 2983 y Ft(i)1645 2971 y Fs(j)k Fv(I)1727 2983 y Ft(i)1778 2971 y FD(=)g(1])18 b Fs(\001)g FD(Prob)o(\()p Fv(I)2235 2983 y Ft(i)2286 2971 y FD(=)23 b(1\))g(=)2598 2867 y Ft(n)2558 2892 y Fu(X)2565 3069 y Ft(i)p FC(=1)2692 2971 y Fv(p)2734 2983 y Ft(i)2762 2971 y FD(E[)o Fv(C)2900 2983 y Ft(i)2951 2971 y Fs(j)g Fv(I)3033 2983 y Ft(i)3084 2971 y FD(=)g(1])p Fv(:)648 3249 y FD(If)28 b(w)n(e)f(could)f(sho)n(w)h(that)1662 3371 y SDict begin H.S end 1662 3371 a 1662 3371 a SDict begin 12 H.A end 1662 3371 a 1662 3371 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.55) cvn /DEST pdfmark end 1662 3371 a FD(E[)o Fv(C)1800 3383 y Ft(i)1828 3371 y Fv(;)14 b(I)1901 3383 y Ft(i)1952 3371 y FD(=)23 b(1])f Fs(\024)h Fv(\014)995 b FD(\(55\))523 3543 y(for)26 b(all)f Fv(i)d Fs(2)i(f)p FD(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(n)p Fs(g)26 b FD(then)h(it)f(w)n(ould)f(follo)n(w)g(that)2210 3481 y Fu(P)2297 3502 y Ft(n)2297 3568 y(i)p FC(=1)2423 3543 y Fv(p)2465 3555 y Ft(i)2515 3543 y Fs(\025)e Fv(n=\014)t FD(,)k(giving)c(us)k(a)g(lo)n(w)n(er)523 3643 y(b)r(ound)h(on)f(the)h (sum)f(of)h(the)g Fv(p)1493 3655 y Ft(i)1520 3643 y FD('s.)648 3750 y(In)h(fact)g(w)n(e)f(sho)n(w)g(\(Lemma)1578 3750 y SDict begin H.S end 1578 3750 a Black FD(18)p Black 1661 3697 a SDict begin H.R end 1661 3697 a 1661 3750 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.18) cvn H.B /ANN pdfmark end 1661 3750 a FD(\))h(that)h(the)f(b)r(ound)g(\()2337 3750 y SDict begin H.S end 2337 3750 a Black FD(55)p Black 2420 3697 a SDict begin H.R end 2420 3697 a 2420 3750 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.55) cvn H.B /ANN pdfmark end 2420 3750 a FD(\))h(holds)d(for)i Fv(i)24 b FD(=)h(1)p Fv(;)14 b(:)g(:)g(:)f(;)h Fs(b)p Fv(n=)p FD(4)o Fs(c)523 3850 y FD(with)1262 3990 y SDict begin H.S end 1262 3990 a 1262 3990 a SDict begin 12 H.A end 1262 3990 a 1262 3990 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.56) cvn /DEST pdfmark end 1262 3990 a Fv(\014)28 b FD(=)1438 3934 y(1)p 1434 3971 50 4 v 1434 4047 a Fv(n)1540 3990 y FD(+)46 b(1)18 b(+)2156 3934 y Fv(\015)5 b(n)p 1804 3971 802 4 v 1804 4080 a FD(1)18 b Fs(\000)1957 4047 y FC(4)p 1957 4061 34 4 v 1957 4108 a(3)2000 3988 y Fu(q)p 2083 3988 523 4 v 2093 4047 a FC(1)p 2093 4061 34 4 v 2093 4108 a(2)2150 4080 y FD(ln)2288 4047 y FC(1)p 2243 4061 124 4 v 2243 4108 a(1)p Fq(\000)p Ft(\015)2394 4080 y FD(+)2487 4047 y FC(ln)11 b Ft(n)p 2487 4061 108 4 v 2521 4108 a(n)2643 3990 y Fv(:)591 b FD(\(56\))648 4280 y(T)-7 b(o)27 b(get)g(an)g(in)n (tuition)e(of)j(wh)n(y)f(the)h(upp)r(er)g(b)r(ound)1092 4512 y SDict begin H.S end 1092 4512 a 1092 4512 a SDict begin 12 H.A end 1092 4512 a 1092 4512 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.57) cvn /DEST pdfmark end 1092 4512 a FD(E[)o Fv(C)1230 4524 y Ft(i)1258 4512 y Fv(;)14 b(I)1331 4524 y Ft(i)1382 4512 y FD(=)23 b(1])g Fs(\024)1659 4456 y FD(1)p 1655 4493 50 4 v 1655 4569 a Fv(n)1761 4512 y FD(+)46 b(1)18 b(+)2377 4456 y Fv(\015)5 b(n)p 2025 4493 802 4 v 2025 4602 a FD(1)18 b Fs(\000)2178 4569 y FC(4)p 2178 4583 34 4 v 2178 4630 a(3)2221 4510 y Fu(q)p 2304 4510 523 4 v 2314 4569 a FC(1)p 2314 4583 34 4 v 2314 4630 a(2)2371 4602 y FD(ln)2509 4569 y FC(1)p 2464 4583 124 4 v 2464 4630 a(1)p Fq(\000)p Ft(\015)2615 4602 y FD(+)2708 4569 y FC(ln)11 b Ft(n)p 2708 4583 108 4 v 2742 4630 a(n)3257 4512 y FD(\(57\))523 4825 y(holds)39 b(for)g Fv(i)k Fs(\024)h(b)p Fv(n=)p FD(4)o Fs(c)o FD(,)d(w)n(e)e(note) h(that)g(for)g(small)c Fv(\015)45 b FD(it)40 b(is)e(close)h(to)h(1)26 b(+)g Fv(\015)5 b(n)40 b FD(and)f(w)n(e)523 4924 y(exp)r(ect)26 b(to)g(see)g Fs(\024)c Fv(\015)5 b(n)26 b FD(p)r(oin)n(ts)f(in)g(an)h (in)n(terv)-5 b(al)23 b(of)j(length)f Fv(\015)31 b FD(follo)n(wing)21 b(an)n(y)26 b(sp)r(eci\014c)f(p)r(oin)n(t.)p Black 1922 5173 a(33)p Black eop end end %%Page: 34 34 TeXDict begin HPSdict begin 34 33 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.34) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(Sp)r(eci\014cally)-7 b(,)25 b(note)i(that)h(when)g Fv(\015)g Fs(!)23 b FD(0)k(since)f Fv(n)d(>)g FD(2)2206 377 y Fu(p)p 2288 377 453 4 v 2288 448 a Fv(e)18 b Fs(\001)h FD(ln)o(\(1)p Fv(=\015)5 b FD(\))p Fv(=\015)31 b FD(then)d Fv(n)23 b Fs(!)g(1)28 b FD(and)1414 723 y(lim)1403 775 y Ft(\015)t Fq(!)p FC(0)1271 869 y Ft(n>)1364 800 y Fa(q)p 1429 800 244 3 v 1439 840 a Fo(e)10 b Fb(ln\(1)p Fo(=\015)s Fb(\))p 1439 856 224 3 v 1534 889 a Fo(\015)1697 667 y FD(4)p 1697 704 42 4 v 1697 780 a(3)1748 574 y Fu(s)p 1831 574 624 4 v 1841 667 a FD(1)p 1841 704 42 4 v 1841 780 a(2)1906 723 y(ln)2074 667 y(1)p 1999 704 191 4 v 1999 780 a(1)18 b Fs(\000)g Fv(\015)2218 723 y FD(+)2311 667 y(ln)13 b Fv(n)p 2311 704 133 4 v 2353 780 a(n)2477 723 y FD(=)23 b(0)k Fv(;)523 1066 y FD(so)g(E[)p Fv(C)764 1078 y Ft(i)815 1066 y Fs(j)c Fv(I)897 1078 y Ft(i)948 1066 y FD(=)g(1])f Fs(\031)h FD(1)18 b(+)g Fv(\015)5 b(n)o(:)648 1167 y FD(W)-7 b(e)27 b(pro)n(v)n(e)e(Inequalit)n(y)g(\()1437 1167 y SDict begin H.S end 1437 1167 a Black FD(57)p Black 1520 1114 a SDict begin H.R end 1520 1114 a 1520 1167 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.57) cvn H.B /ANN pdfmark end 1520 1167 a FD(\))i(for)g Fv(i)c FD(=)f(1)p Fv(;)14 b(:)g(:)g(:)f(;)h Fs(b)p Fv(n=)p FD(4)o Fs(c)27 b FD(in)f(t)n(w)n(o)g (steps.)h(Lemma)3084 1167 y SDict begin H.S end 3084 1167 a Black FD(19)p Black 3167 1114 a SDict begin H.R end 3167 1114 a 3167 1167 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.19) cvn H.B /ANN pdfmark end 3167 1167 a 27 w FD(sho)n(ws)523 1267 y(that)h(if)f(the)h(greedy)e(algorithm)e(c)n (ho)r(oses)i Fv(q)1899 1279 y Ft(i)1954 1267 y FD(then)i(with)g(high)e (probabilit)n(y)1336 1533 y SDict begin H.S end 1336 1533 a 1336 1533 a SDict begin 12 H.A end 1336 1533 a 1336 1533 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.58) cvn /DEST pdfmark end 1336 1533 a Fv(X)1405 1545 y Ft(i)1455 1533 y Fs(\024)1553 1477 y Fv(i)18 b Fs(\000)g FD(1)p 1553 1514 172 4 v 1613 1590 a Fv(n)1753 1533 y FD(+)1836 1383 y Fu(s)p 1919 1383 624 4 v 1929 1477 a FD(1)p 1929 1514 42 4 v 1929 1590 a(2)1994 1533 y(ln)2161 1477 y(1)p 2087 1514 191 4 v 2087 1590 a(1)g Fs(\000)g Fv(\015)2306 1533 y FD(+)2399 1477 y(ln)13 b Fv(n)p 2399 1514 133 4 v 2440 1590 a(n)2569 1533 y(:)665 b FD(\(58\))523 1785 y(\(Informally)35 b(this)j(is)f(to)i(sa)n(y)e(that)i Fv(X)1764 1797 y Ft(i)1830 1785 y FD(is)e(concen)n(trated)g(around)h (its)f(mean,)h(whic)n(h)f(is)523 1885 y Fv(i=)p FD(\()p Fv(n)19 b FD(+)g(1\)\).)29 b(Next,)h(Lemma)1463 1885 y SDict begin H.S end 1463 1885 a Black FD(21)p Black 1546 1831 a SDict begin H.R end 1546 1831 a 1546 1885 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.21) cvn H.B /ANN pdfmark end 1546 1885 a 29 w FD(and)f(Lemma)2036 1885 y SDict begin H.S end 2036 1885 a Black FD(22)p Black 2119 1831 a SDict begin H.R end 2119 1831 a 2119 1885 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.22) cvn H.B /ANN pdfmark end 2119 1885 a 30 w FD(imply)d(that)k(if)e(the)i(greedy)e (algorithm)523 1984 y(c)n(ho)r(oses)e Fv(X)888 1996 y Ft(i)943 1984 y FD(then)1314 2118 y SDict begin H.S end 1314 2118 a 1314 2118 a SDict begin 12 H.A end 1314 2118 a 1314 2118 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.59) cvn /DEST pdfmark end 1314 2118 a FD(E[)o Fv(C)1452 2130 y Ft(i)1503 2118 y Fs(j)e Fv(X)1619 2130 y Ft(i)1669 2118 y FD(=)f Fv(x)1804 2130 y Ft(i)1832 2118 y FD(])g(=)g(1)18 b(+)2207 2062 y(1)p 2119 2099 219 4 v 2119 2175 a(1)f Fs(\000)h Fv(x)2308 2187 y Ft(i)2347 2118 y FD(\()p Fv(n)g Fs(\000)g Fv(i)p FD(\))p Fv(:)643 b FD(\(59\))523 2312 y(Substituting)27 b(Equation)e(\()1389 2312 y SDict begin H.S end 1389 2312 a Black FD(58)p Black 1472 2258 a SDict begin H.R end 1472 2258 a 1472 2312 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.58) cvn H.B /ANN pdfmark end 1472 2312 a FD(\))j(in)f(\()1661 2312 y SDict begin H.S end 1661 2312 a Black FD(59)p Black 1745 2258 a SDict begin H.R end 1745 2258 a 1745 2312 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.59) cvn H.B /ANN pdfmark end 1745 2312 a FD(\))h(w)n(e)f(get)g(\()2097 2312 y SDict begin H.S end 2097 2312 a Black FD(57)p Black 2180 2258 a SDict begin H.R end 2180 2258 a 2180 2312 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.57) cvn H.B /ANN pdfmark end 2180 2312 a FD(\).)648 2413 y(The)18 b(claim)e(that)j Fv(n)k FD(=)1353 2351 y Fu(P)1441 2371 y Ft(n)1441 2438 y(i)p FC(=0)1567 2413 y Fv(p)1609 2425 y Ft(i)1636 2413 y FD(E[)p Fv(C)1775 2425 y Ft(i)1826 2413 y Fs(j)g Fv(I)1908 2425 y Ft(i)1959 2413 y FD(=)g(1])18 b(whic)n(h)g(w)n(e)g(men)n(tioned) f(b)r(efore,)h(extends)523 2512 y(\(with)27 b(the)h(same)e(argumen)n (t\))g(to)i(partial)c(sums.)j Fw(I.e.)p FD(,)h(for)f(all)e Fv(k)h Fs(\024)d Fv(n)1521 2779 y(k)j Fs(\024)1719 2676 y Ft(k)1678 2701 y Fu(X)1684 2877 y Ft(i)p FC(=0)1811 2779 y Fv(p)1853 2791 y Ft(i)1881 2779 y FD(E[)o Fv(C)2019 2791 y Ft(i)2071 2779 y Fs(j)d Fv(I)2153 2791 y Ft(i)2204 2779 y FD(=)f(1])28 b Fv(:)523 3042 y FD(Using)39 b(this)h(with)g Fv(k)48 b FD(=)c Fs(b)p Fv(n=)p FD(4)o Fs(c)c FD(and)g(substituting)g (the)g(b)r(ound)h(\()2699 3042 y SDict begin H.S end 2699 3042 a Black FD(55)p Black 2783 2988 a SDict begin H.R end 2783 2988 a 2783 3042 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.55) cvn H.B /ANN pdfmark end 2783 3042 a FD(\))g(whic)n(h)e(w)n(e)h(ha)n(v)n(e)523 3156 y(pro)n(v)n(en)f(for)h (1)k Fs(\024)g Fv(i)g Fs(\024)g(b)p Fv(n=)p FD(4)o Fs(c)p FD(,)c(w)n(e)h(get)f(that)2073 3094 y Fu(P)2160 3114 y Fq(b)q Ft(n=)p FC(4)p Fq(c)2160 3181 y Ft(i)p FC(=0)2347 3156 y Fv(p)2389 3168 y Ft(i)2461 3156 y Fs(\025)k FD(1)p Fv(=\014)22 b Fs(\001)c(b)p Fv(n=)p FD(4)o Fs(c)p FD(.)41 b(Let)f Fv(A)3258 3168 y FC(1)3340 3156 y FD(=)523 3256 y Fs(f)p Fv(X)634 3268 y FC(1)671 3256 y Fv(;)14 b(:)g(:)g(:)f(;)h(X) 924 3271 y Fq(b)p Ft(n=)p FC(4)p Fq(c)1097 3256 y Fs(g)p FD(,)33 b(w)n(e)g(ha)n(v)n(e)f(that)h(the)h(exp)r(ected)g(size)e(of)h (the)g(maximal)c(indep)r(enden)n(t)523 3355 y(set)f(in)f Fv(A)g FD(is)g(at)g(least)g Fs(b)o Fv(n=)p FD(4)o Fs(c)p Fv(=\014)t FD(.)648 3456 y(In)20 b(Lemma)1033 3456 y SDict begin H.S end 1033 3456 a Black FD(23)p Black 1116 3403 a SDict begin H.R end 1116 3403 a 1116 3456 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.23) cvn H.B /ANN pdfmark end 1116 3456 a 21 w FD(w)n(e)g(sho)n(w)f(that)i(for)f(an)n (y)f(0)k Fs(\024)g Fv(i)f(<)h FD(4)d(the)h(exp)r(ected)f(size)f(of)i (the)g(maximal)523 3556 y Fv(\015)5 b FD(-indep)r(enden)n(t)28 b(set)g(in)g(the)h(p)r(oin)n(t)f(sets)g Fv(A)1885 3568 y Ft(i)1938 3556 y FD(=)c Fs(f)p Fv(X)2138 3571 y Ft(i)p Fq(b)p Ft(n=)p FC(4)p Fq(c)q FC(+1)2418 3556 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)2672 3571 y FC(\()p Ft(i)p FC(+1\))p Fq(b)p Ft(n=)p FC(4)p Fq(c)3004 3556 y Fs(g)28 b FD(are)g(equal)523 3655 y(\(and)g(th)n(us)f(at)h(least)e Fs(b)p Fv(n=)p FD(4)o Fs(c)p Fv(=\014)t FD(\).)648 3756 y(It)21 b(therefore)f(follo)n (ws)e(that)k(in)e(the)i(union)e Fv(A)2026 3768 y FC(1)2069 3756 y Fs([)5 b Fv(A)2191 3768 y FC(2)2234 3756 y Fs([)g Fv(A)2356 3768 y FC(3)2400 3756 y Fs([)g Fv(A)2522 3768 y FC(4)2582 3756 y FD(the)21 b(size)f(of)h(the)h(maximal)523 3856 y Fv(\015)5 b FD(-indep)r(enden)n(t)36 b(set)g(is)f(at)i(least)e (4)p Fs(b)o Fv(n=)p FD(4)o Fs(c)p Fv(=\014)28 b Fs(\000)c FD(3.)36 b(Ergo,)f(the)h(size)g(of)g(the)h(maximal)32 b Fv(\015)5 b FD(-)523 3956 y(indep)r(enden)n(t)27 b(set)g(in)g Fv(q)1256 3968 y FC(1)1293 3956 y Fv(;)14 b(:)g(:)g(:)g(;)g(q)1515 3968 y Ft(n)1587 3956 y FD(is)27 b(at)g(least)f(\()p Fv(n)18 b Fs(\000)f FD(1\))p Fv(=\014)22 b Fs(\000)17 b FD(3)23 b Fs(\025)f Fv(n=\014)g Fs(\000)17 b FD(4.)27 b(As)g(the)h(greedy)523 4055 y(algorithm)i(is)j(optimal)e(\(Lemma)1644 4055 y SDict begin H.S end 1644 4055 a Black FD(12)p Black 1727 4002 a SDict begin H.R end 1727 4002 a 1727 4055 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.12) cvn H.B /ANN pdfmark end 1727 4055 a FD(\))j(it)f(follo)n(ws)e(that)k(the)f (greedy)f(algorithm)d(giv)n(es)h(a)523 4155 y Fv(\015)5 b FD(-indep)r(enden)n(t)27 b(set)g(of)h(size)e(at)i(least)e Fv(n=\014)c Fs(\000)c FD(4.)648 4256 y(The)27 b(theorem)f(follo)n(ws)f (b)n(y)i(sho)n(wing)f(that)i(for)f(\\large")d(n,)1419 4401 y Fv(n)p 1418 4438 52 4 v 1418 4514 a(\014)1498 4458 y Fs(\000)18 b Fv(O)r FD(\(1\))24 b Fs(\025)1969 4401 y Fv(n)p 1873 4438 241 4 v 1873 4514 a FD(1)18 b(+)g Fv(n\015)2124 4458 y FD(\(1)g Fs(\000)g Fv(\013)p FD(\()p Fv(\015)5 b FD(\)\))p Fv(:)523 4685 y SDict begin H.S end 523 4685 a 523 4685 a SDict begin 12 H.A end 523 4685 a 523 4685 a SDict begin [/View [/XYZ H.V]/Dest (subsection.A.3.2) cvn /DEST pdfmark end 523 4685 a 67 x Fr(C.2)95 b(F)-8 b(ull)31 b(Pro)s(of)g(of)h(theorem)1700 4752 y SDict begin H.S end 1700 4752 a Black Fr(3)p Black 1748 4699 a SDict begin H.R end 1748 4699 a 1748 4752 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 1748 4752 a Black 523 4924 a Fp(Pr)l(o)l(of.)p Black 43 w FD(First)27 b(let)g(mak)n(e)e(the)j(follo)n(wing)c (de\014nitions:)p Black 1922 5173 a(34)p Black eop end end %%Page: 35 35 TeXDict begin HPSdict begin 35 34 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.35) cvn /DEST pdfmark end 515 125 a Black Black Black 575 448 a Fr({)p Black 41 w FD(De\014ne)34 b Fv(Z)k FD(b)r(e)c(the)f(set)g Fs(f)p Fv(z)1505 460 y FC(1)1542 448 y Fv(;)14 b(z)1618 460 y FC(2)1654 448 y Fv(;)g(:)g(:)g(:)g(;)g(z)1878 460 y Ft(n)1923 448 y Fs(g)32 b FD(where)h(eac)n(h)f Fv(z)2474 460 y Ft(i)2534 448 y FD(is)g(sampled)f(uniformly)f(in)664 548 y([0)p Fv(;)14 b FD(1\).)p Black 575 648 a Fr({)p Black 41 w FD(De\014ne)26 b(the)g(random)e(v)-5 b(ariable)23 b Fv(X)1742 660 y Ft(i)1769 648 y FD(,)j(1)c Fs(\024)h Fv(i)g Fs(\024)f Fv(n)k FD(to)f(b)r(e)h(the)h Fv(i)p FD('th)e(smallest)e(p)r(oin)n(t)h (in)h Fv(Z)6 b FD(.)p Black 575 748 a Fr({)p Black 41 w FD(De\014ne)40 b(the)g(random)d(v)-5 b(ariable)37 b Fv(C)1787 760 y Ft(i)1854 748 y FD(to)j(b)r(e)f(the)h(n)n(um)n(b)r(er)e (of)i Fv(X)2737 760 y Ft(j)2771 748 y FD('s)g(that)f(lie)f(in)h(the)664 847 y(in)n(terv)-5 b(al)31 b([)p Fv(X)1062 859 y Ft(i)1089 847 y Fv(;)14 b(X)1195 859 y Ft(i)1245 847 y FD(+)21 b Fv(\015)5 b FD(\).)34 b(Note)f(that)g(at)g(most)f(one)h(of)g(the)h Fv(X)2651 859 y Ft(j)2718 847 y Fs(2)f FD([)p Fv(X)2898 859 y Ft(i)2925 847 y Fv(;)14 b(X)3031 859 y Ft(i)3081 847 y FD(+)21 b Fv(\015)5 b FD(\))33 b(can)664 947 y(b)r(elong)26 b(to)i(a)f Fv(\015)5 b FD(-indep)r(enden)n(t)27 b(set.)p Black 575 1047 a Fr({)p Black 41 w FD(Giv)n(en)39 b(an)n(y)h(set)g Fv(S)46 b FD(of)40 b(p)r(oin)n(ts)f(w)n(e)h(de\014ne)h(the)g(follo)n (wing)36 b Fp(gr)l(e)l(e)l(dy)42 b(algorithm)49 b FD(that)664 1147 y(constructs)23 b(a)g Fv(\015)5 b FD(-indep)r(enden)n(t)22 b(set.)i(The)f(greedy)f(algorithm)d(initializes)f(the)24 b(set)f(with)664 1246 y(the)h(smallest)c(p)r(oin)n(t)i(in)h Fv(S)28 b FD(and)23 b(then)h(tra)n(v)n(erses)d(the)i(remaining)d(p)r (oin)n(ts)i(in)g(increasing)664 1346 y(order)i(and)h(adds)f(the)i(p)r (oin)n(t)e Fv(x)i FD(to)f(the)g Fv(\015)5 b FD(-indep)r(enden)n(t)24 b(set)h(if)g Fv(x)h FD(is)e(larger)e(b)n(y)i(at)h(least)664 1446 y Fv(\015)j FD(from)22 b(an)n(y)g(p)r(oin)n(t)g(previously)e (added.)j(W)-7 b(e)24 b(denote)f(b)n(y)g Fv(G)p FD(\()p Fv(S;)14 b(\015)5 b FD(\))23 b(the)h Fv(\015)5 b FD(-indep)r(enden)n(t) 664 1545 y(set)39 b(computed)g(b)n(y)g(applying)d(greedy)i(algorithm)e (to)j Fv(S)5 b FD(.)39 b(Note)g(that)g Fs(j)q Fv(G)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))p Fs(j)42 b FD(=)664 1645 y Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))28 b(\(Lemma)d(\()1332 1645 y SDict begin H.S end 1332 1645 a Black FD(12)p Black 1416 1591 a SDict begin H.R end 1416 1591 a 1416 1645 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.12) cvn H.B /ANN pdfmark end 1416 1645 a FD(\)\).)p Black 575 1745 a Fr({)p Black 41 w FD(Let)32 b Fv(G)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))31 b(b)r(e)h(a)f(random)e (v)-5 b(ariable)28 b(whic)n(h)j(is)f(the)i Fv(\015)5 b FD(-indep)r(enden)n(t)30 b(set)h(obtained)664 1844 y(b)n(y)c(the)h(greedy)f(algorithm)c(when)28 b(applied)e(to)h(the)h (set)g Fv(Z)6 b FD(.)p Black 575 1944 a Fr({)p Black 41 w FD(Let)22 b Fv(I)843 1956 y Ft(i)893 1944 y FD(b)r(e)h(a)e(random) f(v)-5 b(ariable)19 b(with)j(binary)e(v)-5 b(alues)21 b(where)g Fv(I)2620 1956 y Ft(i)2671 1944 y FD(=)i(1)e(i\013)h Fv(X)2985 1956 y Ft(i)3035 1944 y Fs(2)i Fv(G)p FD(\()p Fv(Z)q(;)14 b(d)p FD(\),)664 2044 y(and)28 b(de\014ne)1700 2144 y Fv(p)1742 2156 y Ft(i)1792 2144 y FD(=)23 b(Prob)n([)14 b Fv(I)2129 2156 y Ft(i)2180 2144 y FD(=)23 b(1)14 b(])o Fv(:)648 2311 y FD(F)-7 b(or)27 b Fv(n)22 b Fs(\024)h FD(2)999 2240 y Fu(p)p 1082 2240 522 4 v 71 x Fv(e)14 b FD(ln)e(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g FD(1)p Fv(=\015)32 b FD(the)c(statemen)n(t)e(follo)n(ws)f(from)h(Lemma)2924 2311 y SDict begin H.S end 2924 2311 a Black FD(13)p Black 3007 2258 a SDict begin H.R end 3007 2258 a 3007 2311 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.13) cvn H.B /ANN pdfmark end 3007 2311 a FD(.)648 2423 y(So)e(from)g(here)g(to)h (the)g(end)g(of)g(the)g(pro)r(of)f(w)n(e)g(assume)g(that)h Fv(n)e(>)f FD(2)2753 2352 y Fu(p)p 2836 2352 522 4 v 71 x Fv(e)14 b FD(ln)e(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g FD(1)p Fv(=\015)29 b FD(.)648 2523 y(Let)e Fv(A)858 2535 y Ft(i)886 2523 y FD(,)h(1)23 b Fs(\024)f Fv(i)h(<)f FD(4,)28 b(b)r(e)g(the)g(set)f Fs(f)p Fv(X)1817 2538 y FC(1+)p Ft(i)p Fq(b)q Ft(n=)p FC(4)p Fq(c)2097 2523 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)2351 2538 y FC(\()p Ft(i)p FC(+1\))p Fq(b)p Ft(n=)p FC(4)q Fq(c)2683 2523 y Fs(g)p FD(.)648 2623 y(Note)27 b(that)h(for)f(an)n(y)g(1)c Fs(\024)f Fv(i)h(<)f FD(4)1011 2806 y(max)12 b(\()q Fv(A)1274 2818 y Ft(i)1302 2806 y FD(\))23 b(=)g Fv(X)1514 2834 y FC(\()p Ft(i)p FC(+1\))1673 2840 y Fs(b)1720 2811 y Fo(n)p 1720 2820 37 3 v 1724 2854 a Fb(4)1766 2840 y Fs(c)1830 2806 y(\024)g Fv(X)1987 2834 y FC(1+\()p Ft(i)p FC(+1\))2230 2840 y Fs(b)2277 2811 y Fo(n)p 2277 2820 V 2281 2854 a Fb(4)2324 2840 y Fs(c)2388 2806 y FD(=)f(min)13 b(\()p Fv(A)2722 2818 y Ft(i)p FC(+1)2834 2806 y FD(\))28 b Fv(:)648 3010 y FD(It)g(is)e(easy)h(to)g(v)n(erify)f(that)990 3268 y(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)g FD(E)1540 3126 y Fu(")1588 3268 y Fv(m)1661 3126 y Fu( )1764 3164 y FC(3)1734 3189 y Fu([)1726 3366 y Ft(i)p FC(=0)1848 3268 y Fv(A)1910 3280 y Ft(i)1938 3268 y Fv(;)14 b(\015)2023 3126 y Fu(!)o(#)2159 3268 y Fs(\025)2291 3164 y FC(3)2247 3189 y Fu(X)2253 3366 y Ft(i)p FC(=0)2381 3268 y FD(E[)o Fv(m)p FD(\()p Fv(A)2627 3280 y Ft(i)2656 3268 y Fv(;)g(\015)5 b FD(\)])18 b Fs(\000)g FD(3)523 3529 y(By)26 b(Lemma)947 3529 y SDict begin H.S end 947 3529 a Black FD(23)p Black 1030 3475 a SDict begin H.R end 1030 3475 a 1030 3529 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.23) cvn H.B /ANN pdfmark end 1030 3529 a FD(,)h(E[)o Fv(m)p FD(\()p Fv(A)1326 3541 y Ft(i)1354 3529 y Fv(;)14 b(\015)5 b FD(\)])23 b(=)g(E[)o Fv(m)p FD(\()p Fv(A)1851 3541 y FC(0)1889 3529 y Fv(;)14 b(\015)5 b FD(\)])26 b(for)g Fv(i)d FD(=)g(1)p Fv(;)14 b FD(2)p Fv(;)g FD(3.)24 b(By)i(Lemma)2992 3529 y SDict begin H.S end 2992 3529 a Black FD(12)p Black 3075 3475 a SDict begin H.R end 3075 3475 a 3075 3529 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.12) cvn H.B /ANN pdfmark end 3075 3529 a FD(,)h Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))523 3628 y(is)34 b(the)i(size)e(of)i(the)g(indep)r(enden)n(t)f (set)h(pic)n(k)n(ed)d(b)n(y)j(the)f(greedy)g(algorithm)c(and)k(b)n(y)h (the)523 3738 y(de\014nition)26 b(of)i(the)g Fv(p)1172 3750 y Ft(i)1199 3738 y FD('s)f(w)n(e)h(ha)n(v)n(e)e(that)i(E[)p Fv(m)p FD(\()p Fv(A)2023 3750 y FC(0)2060 3738 y Fv(;)14 b(\015)5 b FD(\)])23 b(=)2311 3676 y Fu(P)2398 3696 y Fq(b)q Ft(n=)p FC(4)p Fq(c)2398 3763 y Ft(i)p FC(=1)2585 3738 y Fv(p)2627 3750 y Ft(i)2655 3738 y FD(.)28 b(So)f(w)n(e)g(get) 1444 4018 y SDict begin H.S end 1444 4018 a 1444 4018 a SDict begin 12 H.A end 1444 4018 a 1444 4018 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.60) cvn /DEST pdfmark end 1444 4018 a FD(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)g FD(4)18 b Fs(\001)2038 3907 y Fq(b)q Ft(n=)p FC(4)p Fq(c)2063 3939 y Fu(X)2069 4116 y Ft(i)p FC(=1)2221 4018 y Fv(p)2263 4030 y Ft(i)2309 4018 y Fs(\000)g FD(3)27 b Fv(:)773 b FD(\(60\))648 4278 y(F)-7 b(or)27 b(simplicit)n(y)22 b(lets)27 b Fv(r)r FD(\()p Fv(\015)5 b FD(\))28 b(b)r(e)g(as)f(de\014ne)h(is)f(Lemma)2342 4278 y SDict begin H.S end 2342 4278 a Black FD(16)p Black 2425 4225 a SDict begin H.R end 2425 4225 a 2425 4278 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.16) cvn H.B /ANN pdfmark end 2425 4278 a 819 4549 a Fv(r)r FD(\()p Fv(\015)5 b FD(\))24 b(=)f Fv(k)1128 4407 y Fu( )1194 4398 y(s)p 1277 4398 1716 4 v 1287 4493 a FD(1)p 1287 4530 42 4 v 1287 4606 a(2)1352 4549 y(ln)1519 4493 y(1)p 1445 4530 191 4 v 1445 4606 a(1)18 b Fs(\000)g Fv(\015)1664 4549 y FD(+)1757 4493 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p 1757 4530 164 4 v 1818 4606 a(4)1930 4549 y(\()p Fv(\015)19 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))o(\))19 b(+)2446 4493 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln)f(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2446 4530 489 4 v 2669 4606 a(4)2944 4549 y Fv(\015)2992 4407 y Fu(!)3085 4549 y Fv(;)523 4815 y FD(where)31 b Fv(k)i Fs(\031)c FD(2)p Fv(:)p FD(490795)f(is)j(as)g(de\014ned)g(in)g(Lemma) 2167 4815 y SDict begin H.S end 2167 4815 a Black FD(16)p Black 2250 4762 a SDict begin H.R end 2250 4762 a 2250 4815 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.16) cvn H.B /ANN pdfmark end 2250 4815 a 31 w FD(and)h Fv(\014)t FD(\()p Fv(\015)5 b FD(\))30 b(=)f(1)21 b(+)3030 4783 y FC(2)p 2892 4797 310 4 v 2892 4844 a(ln)11 b(\(1)p Ft(=\015)t FC(\))p Fq(\000)p FC(1)3243 4815 y FD(is)31 b(as)523 4924 y(de\014ned)d(in)f(Lemma)1202 4924 y SDict begin H.S end 1202 4924 a Black FD(27)p Black 1285 4871 a SDict begin H.R end 1285 4871 a 1285 4924 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.27) cvn H.B /ANN pdfmark end 1285 4924 a FD(.)p Black 1922 5173 a(35)p Black eop end end %%Page: 36 36 TeXDict begin HPSdict begin 36 35 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.36) cvn /DEST pdfmark end 515 125 a Black Black 648 451 a FD(Substituting)27 b(the)i(lo)n(w)n(er)d(b)r (ound)j(on)1860 388 y Fu(P)1947 409 y Fq(b)q Ft(n=)p FC(4)p Fq(c)1947 476 y Ft(i)p FC(=1)2134 451 y Fv(p)2176 463 y Ft(i)2233 451 y FD(giv)n(en)d(b)n(y)j(Lemma)2864 451 y SDict begin H.S end 2864 451 a Black FD(16)p Black 2947 397 a SDict begin H.R end 2947 397 a 2947 451 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.16) cvn H.B /ANN pdfmark end 2947 451 a 29 w FD(in)f(Equation)523 550 y(\()555 550 y SDict begin H.S end 555 550 a Black FD(60)p Black 638 497 a SDict begin H.R end 638 497 a 638 550 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.60) cvn H.B /ANN pdfmark end 638 550 a FD(\))g(w)n(e)g(get)1051 906 y(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])24 b Fs(\025)e FD(4)c Fs(\001)1656 850 y Fv(n)p 1656 887 50 4 v 1660 963 a FD(4)1734 906 y Fs(\001)1885 850 y FD(1)p 1786 887 241 4 v 1786 963 a(1)g(+)g Fv(\015)5 b(n)2036 789 y Fu(\022)2097 906 y FD(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)2508 850 y FD(4)p 2503 887 50 4 v 2503 963 a Fv(n)2563 789 y Fu(\023)2643 906 y Fs(\000)f FD(3)1457 1139 y(=)1650 1083 y Fv(n)p 1554 1120 241 4 v 1554 1196 a FD(1)g(+)g Fv(\015)5 b(n)1805 1022 y Fu(\022)1866 1139 y FD(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)2276 1083 y FD(4)p 2272 1120 50 4 v 2272 1196 a Fv(n)2350 1139 y Fs(\000)g FD(3)e Fs(\001)2545 1083 y FD(1)h(+)g Fv(\015)5 b(n)p 2545 1120 241 4 v 2640 1196 a(n)2795 1022 y Fu(\023)1457 1371 y FD(=)1650 1315 y Fv(n)p 1554 1352 V 1554 1428 a FD(1)18 b(+)g Fv(\015)5 b(n)1805 1254 y Fu(\022)1866 1371 y FD(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)2276 1315 y FD(4)p 2272 1352 50 4 v 2272 1428 a Fv(n)2350 1371 y Fs(\000)g FD(3)2476 1254 y Fu(\022)2550 1315 y FD(1)p 2546 1352 V 2546 1428 a Fv(n)2624 1371 y FD(+)f Fv(\015)2755 1254 y Fu(\023\023)1457 1604 y FD(=)1650 1547 y Fv(n)p 1554 1585 241 4 v 1554 1661 a FD(1)g(+)g Fv(\015)5 b(n)1805 1487 y Fu(\022)1866 1604 y FD(1)18 b Fs(\000)2009 1487 y Fu(\022)2070 1604 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b(+)2338 1547 y(7)p 2333 1585 50 4 v 2333 1661 a Fv(n)2412 1604 y FD(+)f(3)p Fv(\015)2585 1487 y Fu(\023)o(\023)648 1896 y FD(Since)26 b Fv(n)d(>)g FD(2)1067 1825 y Fu(p)p 1149 1825 522 4 v 1149 1896 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g FD(1)p Fv(=\015)31 b FD(then)857 2163 y SDict begin H.S end 857 2163 a 857 2163 a SDict begin 12 H.A end 857 2163 a 857 2163 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.61) cvn /DEST pdfmark end 857 2163 a FD(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])24 b Fs(\025)1456 2107 y Fv(n)p 1360 2144 241 4 v 1360 2220 a FD(1)18 b(+)g Fv(\015)5 b(n)1611 2046 y Fu(\022)1672 2163 y FD(1)18 b Fs(\000)1815 2046 y Fu(\022)1876 2163 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b(+)2144 2107 y(7)p 2139 2144 50 4 v 2139 2220 a Fv(n)2218 2163 y FD(+)f(3)p Fv(\015)2391 2046 y Fu(\023)o(\023)1263 2420 y Fs(\025)1456 2364 y Fv(n)p 1360 2401 241 4 v 1360 2477 a FD(1)g(+)g Fv(\015)5 b(n)1611 2278 y Fu( )1677 2420 y FD(1)18 b Fs(\000)1820 2278 y Fu( )1885 2420 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))20 b(+)2203 2364 y(7)p 2149 2401 150 4 v 2149 2477 a(2)2191 2418 y Fs(p)p 2259 2418 39 4 v 2259 2477 a Fv(e)2326 2420 y Fs(\001)2500 2308 y(p)p 2569 2308 48 4 v 52 x Fv(\015)p 2378 2401 362 4 v 2378 2418 a Fu(p)p 2461 2418 279 4 v 71 x FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2768 2420 y(+)18 b(3)p Fv(\015)2941 2278 y Fu(!)n(!)p Black 3257 2420 a FD(\(61\))p Black 648 2747 a(No)n(w)23 b(w)n(e)h(upp)r(er)h (b)r(ound)f(the)h(terms)e Fv(r)r FD(\()p Fv(\015)5 b FD(\),)2059 2714 y FC(7)p 2016 2728 120 4 v 2016 2778 a(2)2049 2736 y Fq(p)p 2104 2736 32 3 v 42 x Ft(e)2256 2666 y Fq(p)p 2310 2666 39 3 v 2310 2702 a Ft(\015)p 2156 2727 294 4 v 2156 2737 a Fs(p)p 2225 2737 225 4 v 57 x FC(ln)10 b(\(1)p Ft(=\015)t FC(\))2459 2747 y FD(,)24 b(and)g(3)p Fv(\015)29 b FD(for)24 b(0)e Fv(<)h(\015)28 b(<)22 b(t)p FD(.)j(W)-7 b(e)523 2888 y(start)27 b(with)g Fv(r)r FD(\()p Fv(\015)5 b FD(\).)699 3188 y Fv(r)r FD(\()p Fv(\015)g FD(\))24 b(=)f Fv(k)1008 3036 y Fu(s)p 1091 3036 1716 4 v 1101 3132 a FD(1)p 1101 3169 42 4 v 1101 3245 a(2)1166 3188 y(ln)1334 3132 y(1)p 1259 3169 191 4 v 1259 3245 a(1)18 b Fs(\000)g Fv(\015)1478 3188 y FD(+)1571 3132 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p 1571 3169 164 4 v 1632 3245 a(4)1744 3188 y(\()p Fv(\015)19 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))o(\))19 b(+)2260 3132 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln)f(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2260 3169 489 4 v 2484 3245 a(4)2758 3188 y Fv(\015)874 3471 y FD(=)962 3396 y Fu(p)p 1045 3396 340 4 v 75 x Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g Fv(k)1490 3322 y Fu(s)p 1573 3322 1547 4 v 1583 3415 a FD(1)p 1583 3452 42 4 v 1583 3528 a(2)1653 3471 y Fs(\001)1705 3415 y FD(log)o(\(1)p Fv(=)p FD(\(1)f Fs(\000)i Fv(\015)5 b FD(\)\))p 1705 3452 510 4 v 1790 3528 a Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2243 3471 y(+)2336 3415 y Fv(\014)t FD(\()p Fv(\015)g FD(\))p 2336 3452 164 4 v 2397 3528 a(4)2528 3471 y(+)2621 3415 y Fv(\014)t FD(\()p Fv(\015)g FD(\))14 b(ln)f(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2621 3452 489 4 v 2698 3528 a(4)14 b(ln)f(\(1)p Fv(=\015)5 b FD(\))874 3759 y(=)962 3684 y Fu(p)p 1045 3684 340 4 v 75 x Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g Fv(k)1490 3607 y Fu(s)p 1573 3607 1656 4 v 1583 3703 a FD(1)p 1583 3740 42 4 v 1583 3816 a(2)1653 3759 y Fs(\001)1705 3703 y FD(log)o(\(1)p Fv(=)p FD(\(1)f Fs(\000)i Fv(\015)5 b FD(\))2183 3679 y FC(1)p Ft(=\015)2292 3703 y FD(\))p 1705 3740 620 4 v 1875 3816 a(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2353 3759 y(+)2446 3703 y Fv(\014)t FD(\()p Fv(\015)g FD(\))p 2446 3740 164 4 v 2507 3816 a(4)2638 3759 y(+)2731 3703 y Fv(\014)t FD(\()p Fv(\015)g FD(\))14 b(ln)f(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2731 3740 489 4 v 2808 3816 a(4)14 b(ln)f(\(1)p Fv(=\015)5 b FD(\))874 4045 y Fs(\024)962 3970 y Fu(p)p 1045 3970 340 4 v 75 x Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g Fv(k)1490 3893 y Fu(s)p 1573 3893 1572 4 v 1583 3989 a FD(1)p 1583 4026 42 4 v 1583 4102 a(2)1653 4045 y Fs(\001)1705 3989 y FD(log)o(\(1)p Fv(=)p FD(\(1)f Fs(\000)i Fv(t)p FD(\))2165 3965 y FC(1)p Ft(=t)2261 3989 y FD(\))p 1705 4026 589 4 v 1869 4102 a(ln)13 b(\(1)p Fv(=t)p FD(\))2322 4045 y(+)2415 3989 y Fv(\014)t FD(\()p Fv(t)p FD(\))p 2415 4026 146 4 v 2467 4102 a(4)2589 4045 y(+)2682 3989 y Fv(\014)t FD(\()p Fv(t)p FD(\))h(ln)g(\()p Fv(\014)t FD(\()p Fv(t)p FD(\)\))p 2682 4026 454 4 v 2751 4102 a(4)g(ln)e(\(1)p Fv(=t)p FD(\))3173 4045 y Fv(;)523 4334 y FD(where)37 b(the)h(last)e(inequalit)n(y)e(follo)n(ws)h (since)h(the)i(functions)2610 4301 y FC(1)p 2515 4315 225 4 v 2515 4362 a(ln)10 b(\(1)p Ft(=\015)t FC(\))2749 4334 y FD(,)37 b(ln)13 b(\(1)p Fv(=)p FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))3263 4303 y FC(1)p Ft(=\015)3373 4334 y FD(\))523 4447 y(and)27 b Fv(\014)t FD(\()p Fv(\015)5 b FD(\))29 b(are)d(monotonic)f(increasing)f(in)j Fv(\015)5 b FD(.)27 b(So)h(w)n(e)f(ha)n(v)n(e)f(that)1585 4671 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))24 b Fs(\024)e Fv(c)1883 4683 y FC(1)1921 4596 y Fu(p)p 2004 4596 340 4 v 75 x Fv(\015)c FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))523 4917 y(where)27 b Fv(c)799 4929 y FC(1)859 4917 y FD(=)c Fv(k)993 4820 y Fu(q)p 1076 4820 1312 4 v 1086 4885 a FC(1)p 1086 4899 34 4 v 1086 4946 a(2)1147 4917 y Fs(\001)1199 4877 y FC(log)q(\(1)p Ft(=)p FC(\(1)p Fq(\000)p Ft(t)p FC(\))1540 4860 y Fb(1)p Fo(=t)1626 4877 y FC(\))p 1199 4898 454 4 v 1320 4946 a(ln)11 b(\(1)p Ft(=t)p FC(\))1680 4917 y FD(+)1773 4877 y Ft(\014)s FC(\()p Ft(t)p FC(\))p 1773 4898 118 4 v 1816 4946 a(4)1919 4917 y FD(+)2012 4877 y Ft(\014)s FC(\()p Ft(t)p FC(\))g(ln)g(\()p Ft(\014)s FC(\()p Ft(t)p FC(\)\))p 2012 4898 366 4 v 2067 4946 a(4)h(ln)f(\(1)p Ft(=t)p FC(\))2411 4917 y Fs(\031)22 b FD(2)p Fv(:)p FD(058664)j(is)h(a)h(constan)n(t.)p Black 1922 5173 a(36)p Black eop end end %%Page: 37 37 TeXDict begin HPSdict begin 37 36 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.37) cvn /DEST pdfmark end 515 125 a Black Black 648 452 a FD(No)n(w)27 b(w)n(e)g(lo)r(ok)f(at)h(the)h (term)1635 419 y FC(7)p 1592 433 120 4 v 1592 484 a(2)1625 442 y Fq(p)p 1679 442 32 3 v 1679 484 a Ft(e)1739 452 y Fs(\001)1891 371 y Fq(p)p 1946 371 39 3 v 36 x Ft(\015)p 1791 433 294 4 v 1791 443 a Fs(p)p 1860 443 225 4 v 56 x FC(ln)11 b(\(1)p Ft(=\015)t FC(\))2094 452 y FD(:)875 640 y(7)p 821 677 150 4 v 821 754 a(2)863 694 y Fs(p)p 932 694 39 4 v 60 x Fv(e)999 696 y Fs(\001)1173 584 y(p)p 1242 584 48 4 v 52 x Fv(\015)p 1051 677 362 4 v 1051 694 a Fu(p)p 1134 694 279 4 v 71 x FD(ln)i(\(1)p Fv(=\015)5 b FD(\))1445 696 y(=)1597 640 y(7)p 1543 677 150 4 v 1543 754 a(2)1585 694 y Fs(p)p 1654 694 39 4 v 60 x Fv(e)1831 640 y FD(1)p 1712 677 279 4 v 1712 753 a(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2000 621 y Fu(p)p 2083 621 340 4 v 75 x Fv(\015)19 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))1445 933 y Fs(\024)1734 877 y FD(7)p 1543 914 424 4 v 1543 990 a(2)1585 931 y Fs(p)p 1654 931 39 4 v 59 x Fv(e)13 b FD(ln)g(\(1)p Fv(=t)p FD(\))1995 933 y Fs(\001)2037 858 y Fu(p)p 2120 858 340 4 v 75 x Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))22 b(=)h Fv(c)2606 945 y FC(2)2643 858 y Fu(p)p 2726 858 V 75 x Fv(\015)c FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))26 b Fv(;)523 1136 y FD(where)d(the)g(last)f (inequalit)n(y)e(follo)n(ws)g(since)i(the)h(function)2462 1104 y FC(1)p 2367 1118 225 4 v 2367 1165 a(ln)10 b(\(1)p Ft(=\015)t FC(\))2601 1136 y FD(,)23 b(is)f(monotonic)e(increas-)523 1258 y(ing)26 b(in)h Fv(\015)5 b FD(,)28 b(and)f Fv(c)1054 1270 y FC(2)1114 1258 y FD(=)1366 1225 y FC(7)p 1212 1239 342 4 v 1212 1290 a(2)1245 1248 y Fq(p)p 1299 1248 32 3 v 1299 1290 a Ft(e)12 b FC(ln)f(\(1)p Ft(=t)p FC(\))1586 1258 y Fs(\031)23 b FD(0)p Fv(:)p FD(620670.)648 1370 y(Similarly)f(for)27 b(the)h(term)e(3)p Fv(\015)1121 1559 y FD(3)p Fv(\015)i FD(=)23 b(3)1364 1448 y Fu(r)p 1446 1448 299 4 v 1571 1503 a Fv(\015)p 1456 1540 279 4 v 1456 1616 a FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))1744 1484 y Fu(p)p 1827 1484 340 4 v 75 x Fv(\015)19 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))1234 1828 y Fs(\024)23 b FD(3)1364 1684 y Fu(s)p 1446 1684 281 4 v 1571 1772 a Fv(t)p 1456 1809 261 4 v 1456 1885 a FD(ln)13 b(\(1)p Fv(=t)p FD(\))1727 1753 y Fu(p)p 1810 1753 340 4 v 75 x Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))22 b(=)h Fv(c)2296 1840 y FC(3)2333 1753 y Fu(p)p 2416 1753 V 75 x Fv(\015)c FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))27 b Fv(;)523 2079 y FD(where)g Fv(c)799 2091 y FC(3)859 2079 y FD(=)c(3)989 1991 y Fu(q)p 1072 1991 231 4 v 1174 2047 a Ft(t)p 1081 2061 211 4 v 1081 2108 a FC(ln)11 b(\(1)p Ft(=t)p FC(\))1325 2079 y Fs(\031)23 b FD(0)p Fv(:)p FD(293353.)648 2204 y(Summing)i(all)g (three)i(term)g(together)g(w)n(e)g(get)g(that)614 2393 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b(+)931 2337 y(7)p 877 2374 150 4 v 877 2451 a(2)919 2391 y Fs(p)p 988 2391 39 4 v 60 x Fv(e)1055 2393 y Fs(\001)1229 2281 y(p)p 1298 2281 48 4 v 52 x Fv(\015)p 1106 2374 362 4 v 1106 2391 a Fu(p)p 1189 2391 279 4 v 71 x FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))1496 2393 y(+)18 b(3)p Fv(\015)27 b Fs(\024)c FD(\()p Fv(c)1847 2405 y FC(1)1903 2393 y FD(+)18 b Fv(c)2022 2405 y FC(2)2077 2393 y FD(+)g Fv(c)2196 2405 y FC(3)2234 2393 y FD(\))2266 2318 y Fu(p)p 2349 2318 340 4 v 75 x Fv(\015)g FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))23 b Fs(\024)f FD(3)2841 2318 y Fu(p)p 2924 2318 V 75 x Fv(\015)c FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))27 b Fv(:)523 2605 y FD(Substituting)g(this)f(b)r(ound)i(bac)n(k)f(in)g (Equation)f(\()2098 2605 y SDict begin H.S end 2098 2605 a Black FD(61)p Black 2181 2551 a SDict begin H.R end 2181 2551 a 2181 2605 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.61) cvn H.B /ANN pdfmark end 2181 2605 a FD(\))i(completes)d(the)j(pro)r(of.)370 b Fs(u)-55 b(t)648 2743 y FD(The)27 b(follo)n(wing)d(lemma)g(is)j(w)n(ell)e(kno)n(wn)i (and)g(can)h(b)r(e)g(pro)n(v)n(ed)e(b)n(y)h(induction)f(on)h Fs(j)p Fv(S)5 b Fs(j)p FD(.)523 2764 y SDict begin H.S end 523 2764 a 523 2764 a SDict begin 12 H.A end 523 2764 a 523 2764 a SDict begin [/View [/XYZ H.V]/Dest (lemma.12) cvn /DEST pdfmark end 523 2764 a Black 117 x Fr(Lemma)30 b(12.)p Black 40 w Fp(F)-6 b(or)30 b(any)g(set)g Fv(S)5 b Fp(,)29 b Fs(j)q Fv(G)p FD(\()p Fv(S;)14 b(\015)5 b FD(\))p Fs(j)23 b FD(=)g Fv(m)p FD(\()p Fv(S;)14 b(\015)5 b FD(\))p Fp(.)648 3019 y FD(The)25 b(follo)n(wing)c(lemma)h(sho)n(ws)i (that)i(when)f(\\)p Fv(n)g FD(is)f(small")e(as)i(a)h(function)g(of)g Fv(\015)5 b FD(,)25 b(Equa-)523 3119 y(tion)18 b(\()717 3119 y SDict begin H.S end 717 3119 a Black FD(5)p Black 759 3065 a SDict begin H.R end 759 3065 a 759 3119 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.4.5) cvn H.B /ANN pdfmark end 759 3119 a FD(\))i(holds;)d(th)n(us)j(pro)n(ving)c (Theorem)1850 3119 y SDict begin H.S end 1850 3119 a Black FD(3)p Black 1892 3065 a SDict begin H.R end 1892 3065 a 1892 3119 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.4.3) cvn H.B /ANN pdfmark end 1892 3119 a 19 w FD(for)j(this)f(case.)g (In)i(the)f(pro)r(of)g(of)g(this)f(lemma)e(w)n(e)523 3218 y(will)26 b(use)i(some)f(arithmetical)d(lemmas)h(whic)n(h)i(are)g (pro)n(v)n(ed)g(later)g(\(Lemmas)3007 3218 y SDict begin H.S end 3007 3218 a Black FD(24)p Black 3090 3165 a SDict begin H.R end 3090 3165 a 3090 3218 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.24) cvn H.B /ANN pdfmark end 3090 3218 a 29 w FD(through)523 3318 y SDict begin H.S end 523 3318 a Black FD(26)p Black 606 3264 a SDict begin H.R end 606 3264 a 606 3318 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.26) cvn H.B /ANN pdfmark end 606 3318 a FD(\))523 3339 y SDict begin H.S end 523 3339 a 523 3339 a SDict begin 12 H.A end 523 3339 a 523 3339 a SDict begin [/View [/XYZ H.V]/Dest (lemma.13) cvn /DEST pdfmark end 523 3339 a Black 117 x Fr(Lemma)j(13.)p Black 40 w Fp(F)-6 b(or)30 b(any)g FD(0)23 b Fv(<)f(\015)28 b Fs(\024)23 b Fv(t)p Fp(,)30 b(if)g Fv(n)23 b Fs(\024)g FD(2)2018 3385 y Fu(p)p 2101 3385 508 4 v 71 x Fv(e)14 b FD(ln)n(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g FD(1)p Fv(=\015)t Fp(,)30 b(then:)1208 3632 y FD(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)1806 3576 y Fv(n)p 1711 3613 241 4 v 1711 3689 a FD(1)18 b(+)g Fv(\015)5 b(n)1961 3540 y Fu(\020)2011 3632 y FD(1)18 b Fs(\000)g FD(3)2196 3557 y Fu(p)p 2278 3557 340 4 v 2278 3632 a Fv(\015)h FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2618 3540 y Fu(\021)2697 3632 y Fv(:)p Black 523 3822 a Fp(Pr)l(o)l(of.)p Black 43 w FD(First)21 b(w)n(e)g(pro)n(v)n(e)f(the)i(lemma)c(for)j Fv(n)i FD(=)g(1)e(\(Since)g(w)n(e)g(w)n(ould)f(lik)n(e)g(to)h(use)h (Lemma)3322 3822 y SDict begin H.S end 3322 3822 a Black FD(26)p Black 3405 3769 a SDict begin H.R end 3405 3769 a 3405 3822 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.26) cvn H.B /ANN pdfmark end 3405 3822 a 523 3922 a FD(that)d(w)n(orks)f(only)f(for)i Fv(n)k Fs(\025)f FD(2\).)d(F)-7 b(or)18 b Fv(n)23 b FD(=)g(1,)c Fv(x)1922 3934 y FC(1)1982 3922 y FD(=)k Fv(z)2109 3934 y FC(1)2165 3922 y FD(is)18 b Fv(\015)5 b FD(-indep)r(enden)n(t)18 b(so)g(E[)p Fv(m)p FD(\()p Fv(Z)q(;)c(\015)5 b FD(\)])23 b(=)f(1)523 4022 y(and)27 b(the)h(righ)n(t)e(hand)i(side)e(of)i(Equation)e(in)g(the)i (statemen)n(t)f(is)f(at)i(most)e(1.)648 4121 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)e(the)j(lemma)d(for)i Fv(n)c Fs(\025)g FD(2.)648 4221 y(F)-7 b(or)27 b Fv(n)22 b Fs(\025)h FD(2,)k(using)g (Lemma)1563 4221 y SDict begin H.S end 1563 4221 a Black FD(15)p Black 1646 4167 a SDict begin H.R end 1646 4167 a 1646 4221 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.15) cvn H.B /ANN pdfmark end 1646 4221 a FD(,)h(w)n(e)f(deriv)n(e)f (that)1363 4367 y SDict begin H.S end 1363 4367 a 1363 4367 a SDict begin 12 H.A end 1363 4367 a 1363 4367 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.62) cvn /DEST pdfmark end 1363 4367 a FD(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)g Fv(n)p FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2161 4333 y Ft(n)p Fq(\000)p FC(1)1768 4492 y Fs(\025)23 b Fv(n)p FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2161 4458 y Ft(n)1768 4646 y FD(=)23 b Fv(n)1906 4554 y Fu(\020)1956 4646 y FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2211 4612 y FC(1)p Ft(=\015)2320 4554 y Fu(\021)2370 4571 y Ft(\015)t(n)p Black 3257 4646 a FD(\(62\))p Black 1768 4800 a Fs(\025)23 b Fv(n)p FD(\(\(1)c Fs(\000)f Fv(\015)5 b FD(\))p Fv(=e)p FD(\))2306 4758 y Ft(\015)t(n)p Black 3257 4800 a FD(\(63\))p Black 1768 4924 a(=)23 b Fv(n)p FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2161 4890 y Ft(\015)t(n)2245 4924 y FD(\(1)p Fv(=e)p FD(\))2432 4890 y Ft(\015)t(n)2542 4924 y Fv(:)p Black 692 w FD(\(64\))p Black Black 1922 5173 a(37)p Black eop end end %%Page: 38 38 TeXDict begin HPSdict begin 38 37 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.38) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(The)41 b(expression)e(in)h(\()1263 448 y SDict begin H.S end 1263 448 a Black FD(62)p Black 1346 395 a SDict begin H.R end 1346 395 a 1346 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.62) cvn H.B /ANN pdfmark end 1346 448 a FD(\))i(is)e(no)h(smaller)c(than)k(the)h (expression)c(in)i(\()2865 448 y SDict begin H.S end 2865 448 a Black FD(63)p Black 2949 395 a SDict begin H.R end 2949 395 a 2949 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.62) cvn H.B /ANN pdfmark end 2949 448 a FD(\))h(since)f(\(1) 27 b Fs(\000)523 548 y Fv(\015)5 b FD(\))603 518 y FC(1)p Ft(=\015)t Fq(\000)p FC(1)821 548 y Fs(\025)22 b FD(1)p Fv(=e)i FD(for)h(an)n(y)f(0)f Fv(<)f(\015)28 b(<)23 b FD(1,)h(see)h(Lemma)2160 548 y SDict begin H.S end 2160 548 a Black FD(24)p Black 2243 494 a SDict begin H.R end 2243 494 a 2243 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.24) cvn H.B /ANN pdfmark end 2243 548 a FD(.)g(W)-7 b(e)26 b(no)n(w)e(giv)n(e)g(a)g(lo)n(w)n(er)f(b)r(ound)j(for)523 648 y(the)i(terms)e(in)h(Equation)f(\()1385 648 y SDict begin H.S end 1385 648 a Black FD(64)p Black 1468 594 a SDict begin H.R end 1468 594 a 1468 648 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.62) cvn H.B /ANN pdfmark end 1468 648 a FD(\).)i(F)-7 b(or)27 b(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))1955 617 y Ft(\015)t(n)2067 648 y FD(w)n(e)27 b(ha)n(v)n(e)1405 843 y SDict begin H.S end 1405 843 a 1405 843 a SDict begin 12 H.A end 1405 843 a 1405 843 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.65) cvn /DEST pdfmark end 1405 843 a FD(\(1)19 b Fs(\000)f Fv(\015)5 b FD(\))1661 808 y Ft(\015)t(n)1768 843 y Fs(\025)22 b FD(\(1)d Fs(\000)f Fv(\015)5 b FD(\))2111 808 y FC(2)2144 752 y Fs(p)p 2213 752 306 4 v 56 x Ft(e\015)15 b FC(ln)c(\(1)p Ft(=\015)t FC(\))p Black 3257 843 a FD(\(65\))p Black 1768 967 a Fs(\025)22 b FD(\(1)d Fs(\000)f Fv(\015)5 b FD(\))2111 933 y FC(2)p Black 3257 967 a FD(\(66\))p Black 1768 1131 a Fs(\025)1965 1075 y FD(1)p 1865 1112 241 4 v 1865 1188 a(1)18 b(+)g Fv(\015)5 b(n)2116 1131 y FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))28 b Fv(:)p Black 835 w FD(\(67\))p Black 523 1359 a(Inequalit)n(y)21 b(\()943 1359 y SDict begin H.S end 943 1359 a Black FD(65)p Black 1026 1305 a SDict begin H.R end 1026 1305 a 1026 1359 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.65) cvn H.B /ANN pdfmark end 1026 1359 a FD(\))j(follo)n(ws)c(since)i Fv(n)h Fs(\024)g FD(2)1752 1288 y Fu(p)p 1834 1288 522 4 v 1834 1359 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)i FD(1)p Fv(=\015)t FD(.)k(Inequalit)n(y)e(\()2822 1359 y SDict begin H.S end 2822 1359 a Black FD(66)p Black 2905 1305 a SDict begin H.R end 2905 1305 a 2905 1359 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.65) cvn H.B /ANN pdfmark end 2905 1359 a FD(\))j(follo)n(ws)c(since)523 1458 y(the)33 b(function)f Fv(x)14 b FD(ln\(1)p Fv(=x)p FD(\))33 b(is)e(smaller)e(than)k(1)p Fv(=e)e FD(for)h(ev)n(ery)f(0)g Fv(<)g(x)p FD(,)j(see)e(Lemma)3162 1458 y SDict begin H.S end 3162 1458 a Black FD(25)p Black 3245 1405 a SDict begin H.R end 3245 1405 a 3245 1458 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.25) cvn H.B /ANN pdfmark end 3245 1458 a FD(.)h(In-)523 1558 y(equalit)n(y)26 b(\()872 1558 y SDict begin H.S end 872 1558 a Black FD(67)p Black 955 1504 a SDict begin H.R end 955 1504 a 955 1558 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.65) cvn H.B /ANN pdfmark end 955 1558 a FD(\))j(follo)n(ws)d(since)h(\(1)19 b Fs(\000)f Fv(\015)5 b FD(\))25 b Fs(\025)f FD(1)p Fv(=)p FD(\(1)18 b(+)g Fv(n\015)5 b FD(\))29 b(for)f Fv(n)c Fs(\025)g FD(2)k(and)h(0)24 b Fv(<)g(\015)29 b(<)24 b FD(1)p Fv(=)p FD(2,)j(see)523 1658 y(Lemma)819 1658 y SDict begin H.S end 819 1658 a Black FD(26)p Black 902 1604 a SDict begin H.R end 902 1604 a 902 1658 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.26) cvn H.B /ANN pdfmark end 902 1658 a FD(.)648 1757 y(F)-7 b(or)25 b(\(1)p Fv(=e)p FD(\))982 1727 y Ft(\015)t(n)1065 1757 y FD(,)h(using)f(the)h (T)-7 b(a)n(ylor)24 b(expansion)g(of)i Fv(e)2247 1727 y Fq(\000)p Ft(x)2366 1757 y FD(around)g(0)f(w)n(e)h(get)g(that)g(for)g (an)n(y)523 1871 y Fv(x)i FD(there)g(exists)e Fv(\022)k FD(s.t.)e Fv(e)1287 1841 y Fq(\000)p Ft(x)1403 1871 y FD(=)23 b(1)18 b Fs(\000)g Fv(x)g FD(+)h(\()p Fv(e)1854 1841 y Fq(\000)p Ft(\022)1943 1871 y FD(\))1985 1838 y Ft(x)2023 1813 y Fb(2)p 1985 1852 71 4 v 2004 1899 a FC(2)2088 1871 y Fs(\025)k FD(1)18 b Fs(\000)g Fv(x)p FD(,)28 b(so)1596 2039 y(\(1)p Fv(=e)p FD(\))1783 2005 y Ft(\015)t(n)1889 2039 y Fs(\025)22 b FD(\(1)d Fs(\000)f Fv(\015)5 b(n)p FD(\))27 b Fv(:)648 2208 y FD(Plugging)d(these)j(lo)n (w)n(er)e(b)r(ounds)j(bac)n(k)f(in)n(to)f(Equation)g(\()2466 2208 y SDict begin H.S end 2466 2208 a Black FD(64)p Black 2549 2154 a SDict begin H.R end 2549 2154 a 2549 2208 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.62) cvn H.B /ANN pdfmark end 2549 2208 a FD(\))i(w)n(e)f(get)866 2376 y SDict begin H.S end 866 2376 a 866 2376 a SDict begin 12 H.A end 866 2376 a 866 2376 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.68) cvn /DEST pdfmark end 866 2376 a FD(E[)o Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)g Fv(n)p FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))1664 2342 y Ft(\015)t(n)1748 2376 y FD(\(1)p Fv(=e)p FD(\))1935 2342 y Ft(\015)t(n)1271 2540 y Fs(\025)23 b Fv(n)1518 2484 y FD(1)p 1419 2521 241 4 v 1419 2597 a(1)18 b(+)g Fv(\015)5 b(n)1669 2540 y FD(\(1)19 b Fs(\000)f Fv(\015)5 b FD(\)\(1)18 b Fs(\000)g Fv(\015)5 b(n)p FD(\))1271 2738 y(=)1464 2682 y Fv(n)p 1369 2719 V 1369 2795 a FD(1)18 b(+)g Fv(\015)5 b(n)1620 2671 y Fu(\000)1658 2738 y FD(1)18 b Fs(\000)g Fv(\015)5 b(n)17 b Fs(\000)h Fv(\015)23 b FD(+)c Fv(\015)2197 2704 y FC(2)2233 2738 y Fv(n)2283 2671 y Fu(\001)1271 2936 y Fs(\025)1464 2880 y Fv(n)p 1369 2917 V 1369 2993 a FD(1)f(+)g Fv(\015)5 b(n)1620 2936 y FD(\(1)18 b Fs(\000)g Fv(\015)5 b(n)18 b Fs(\000)g Fv(\015)5 b FD(\))1271 3138 y Fs(\025)1464 3082 y Fv(n)p 1369 3119 V 1369 3195 a FD(1)18 b(+)g Fv(\015)5 b(n)1620 3046 y Fu(\020)1669 3138 y FD(1)18 b Fs(\000)g Fv(\015)23 b Fs(\001)c FD(2)1962 3063 y Fu(p)p 2044 3063 522 4 v 2044 3138 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)i FD(1)p Fv(=\015)j Fs(\000)c Fv(\015)2715 3046 y Fu(\021)p Black 3257 3138 a FD(\(68\))p Black 1271 3340 a(=)1464 3284 y Fv(n)p 1369 3321 241 4 v 1369 3397 a FD(1)g(+)g Fv(\015)5 b(n)1620 3248 y Fu(\020)1669 3340 y FD(1)18 b Fs(\000)g FD(2)1854 3276 y Fs(p)p 1923 3276 39 4 v 64 x Fv(e)g Fs(\001)2021 3265 y Fu(p)p 2104 3265 340 4 v 75 x Fv(\015)h FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g Fv(\015)2594 3248 y Fu(\021)1271 3567 y FD(=)1464 3511 y Fv(n)p 1369 3548 241 4 v 1369 3624 a FD(1)g(+)g Fv(\015)5 b(n)1620 3450 y Fu(\022)1681 3567 y FD(1)18 b Fs(\000)1824 3492 y Fu(p)p 1907 3492 340 4 v 75 x Fv(\015)g FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2246 3450 y Fu(\022)2307 3567 y FD(2)2349 3503 y Fs(p)p 2418 3503 39 4 v 64 x Fv(e)18 b FD(+)2558 3456 y Fu(r)p 2641 3456 299 4 v 2767 3511 a Fv(\015)p 2651 3548 279 4 v 2651 3624 a FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2939 3450 y Fu(\023\023)1271 3835 y Fs(\025)1464 3779 y Fv(n)p 1369 3816 241 4 v 1369 3892 a FD(1)18 b(+)g Fv(\015)5 b(n)1620 3693 y Fu( )1685 3835 y FD(1)18 b Fs(\000)1828 3760 y Fu(p)p 1911 3760 340 4 v 75 x Fv(\015)h FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2251 3693 y Fu( )2317 3835 y FD(2)2359 3771 y Fs(p)p 2427 3771 39 4 v 2427 3835 a Fv(e)18 b FD(+)2567 3691 y Fu(s)p 2650 3691 281 4 v 2776 3779 a Fv(t)p 2660 3816 261 4 v 2660 3892 a FD(ln)13 b(\(1)p Fv(=t)p FD(\))2931 3693 y Fu(!!)p Black 3257 3835 a FD(\(69\))p Black 1271 4068 a Fs(\025)1464 4011 y Fv(n)p 1369 4048 241 4 v 1369 4124 a FD(1)18 b(+)g Fv(n\015)1620 3975 y Fu(\020)1669 4068 y FD(1)g Fs(\000)g FD(3)1854 3993 y Fu(p)p 1937 3993 340 4 v 75 x Fv(\015)g FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2276 3975 y Fu(\021)2354 4068 y Fv(:)p Black 880 w FD(\(70\))p Black 523 4295 a(Inequalit)n(y)21 b(\()943 4295 y SDict begin H.S end 943 4295 a Black FD(68)p Black 1026 4242 a SDict begin H.R end 1026 4242 a 1026 4295 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.68) cvn H.B /ANN pdfmark end 1026 4295 a FD(\))j(follo)n(ws)c(since)i Fv(n)h Fs(\024)g FD(2)1752 4225 y Fu(p)p 1834 4225 522 4 v 1834 4295 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)i FD(1)p Fv(=\015)t FD(.)k(Inequalit)n(y)e(\()2822 4295 y SDict begin H.S end 2822 4295 a Black FD(69)p Black 2905 4242 a SDict begin H.R end 2905 4242 a 2905 4295 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.68) cvn H.B /ANN pdfmark end 2905 4295 a FD(\))j(follo)n(ws)c(since)523 4395 y(the)31 b(function)1103 4362 y FC(1)p 1008 4376 225 4 v 1008 4424 a(ln)11 b(\(1)p Ft(=\015)t FC(\))1273 4395 y FD(is)30 b(monotonically)25 b(increasing)j(with)i Fv(\015)5 b FD(.)31 b(Inequalit)n(y)d(\()3013 4395 y SDict begin H.S end 3013 4395 a Black FD(70)p Black 3097 4342 a SDict begin H.R end 3097 4342 a 3097 4395 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.68) cvn H.B /ANN pdfmark end 3097 4395 a FD(\))j(follo)n(ws)523 4504 y(b)n(y)c(substituting)g(the) g(v)-5 b(alue)27 b(of)h Fv(t)p FD(.)1744 b Fs(u)-55 b(t)523 4581 y SDict begin H.S end 523 4581 a 523 4581 a SDict begin 12 H.A end 523 4581 a 523 4581 a SDict begin [/View [/XYZ H.V]/Dest (lemma.14) cvn /DEST pdfmark end 523 4581 a Black 83 x Fr(Lemma)30 b(14.)p Black 40 w FD(Prob)o([)14 b Fv(I)1275 4676 y Ft(i)1326 4664 y FD(=)22 b(1)14 b(])23 b Fs(\025)g FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))1858 4634 y Ft(n)p Fq(\000)p FC(1)1988 4664 y Fp(.)p Black 523 4825 a(Pr)l(o)l(of.)p Black 43 w FD(Note)36 b(that)f(for)g(all)e(1) j Fs(\024)f Fv(i;)14 b(k)39 b Fs(\024)c Fv(n)p FD(,)h(the)g(probabilit) n(y)31 b(that)36 b(the)f Fv(k)s FD(th)h(sample)d(of)523 4924 y Fv(Z)40 b FD(is)34 b(the)h Fv(i)p FD(th)f(smallest)d(\(i.e.)j (the)h(probabilit)n(y)c(that)k Fv(X)2339 4936 y Ft(i)2401 4924 y FD(=)f Fv(y)2541 4936 y Ft(k)2581 4924 y FD(\))h(is)f(exactly)f (1)p Fv(=n)p FD(.)g(Also,)p Black 1922 5173 a(38)p Black eop end end %%Page: 39 39 TeXDict begin HPSdict begin 39 38 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.39) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(note)29 b(that)g(for)g(an)n(y)f(1)d Fs(\024)g Fv(k)s(;)14 b(j)30 b Fs(\024)25 b Fv(n)p FD(,)k Fv(k)f Fs(6)p FD(=)d Fv(j)5 b FD(,)29 b(the)h(probabilit)n(y)25 b(that)k Fv(y)2718 460 y Ft(j)2782 448 y FD(falls)e(in)n(to)h(the)h (real)523 548 y(in)n(terv)-5 b(al)25 b([)p Fv(y)887 560 y Ft(k)946 548 y Fs(\000)18 b Fv(\015)5 b(;)14 b(y)1155 560 y Ft(k)1195 548 y FD(\))28 b(is)f(at)g(most)g Fv(\015)5 b FD(.)27 b(It)h(therefore)e(follo)n(ws)f(that)657 792 y(Prob)o([)14 b Fv(I)907 804 y Ft(i)958 792 y FD(=)23 b(1)14 b(])22 b Fs(\025)1275 689 y Ft(n)1235 714 y Fu(X)1235 892 y Ft(k)q FC(=1)1370 792 y FD(Prob)n([)14 b Fv(X)1652 804 y Ft(i)1703 792 y FD(=)22 b Fv(y)1831 804 y Ft(k)1886 792 y FD(])c Fs(\001)h FD(Prob)2145 725 y Fu(\002)2180 792 y FD(for)27 b(all)e Fv(j)j Fs(6)p FD(=)23 b Fv(k)s(;)14 b(y)2696 804 y Ft(j)2753 792 y Fs(62)24 b FD([)p Fv(y)2896 804 y Ft(k)2955 792 y Fs(\000)18 b Fv(\015)5 b(;)14 b(y)3164 804 y Ft(k)3204 792 y FD(\))3236 725 y Fu(\003)1147 1071 y FD(=)1249 1015 y(1)p 1245 1052 50 4 v 1245 1128 a Fv(n)1358 967 y Ft(n)1319 992 y Fu(X)1319 1171 y Ft(k)q FC(=1)1439 1071 y FD(\(1)19 b Fs(\000)f Fv(\015)5 b FD(\))1695 1037 y Ft(n)p Fq(\000)p FC(1)1848 1071 y FD(=)1949 1015 y(1)p 1945 1052 V 1945 1128 a Fv(n)2024 1071 y Fs(\001)18 b Fv(n)p FD(\(1)g Fs(\000)g Fv(\015)5 b FD(\))2370 1037 y Ft(n)p Fq(\000)p FC(1)2523 1071 y FD(=)23 b(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2866 1037 y Ft(n)p Fq(\000)p FC(1)3024 1071 y Fv(:)3350 1312 y Fs(u)-55 b(t)523 1382 y SDict begin H.S end 523 1382 a 523 1382 a SDict begin 12 H.A end 523 1382 a 523 1382 a SDict begin [/View [/XYZ H.V]/Dest (lemma.15) cvn /DEST pdfmark end 523 1382 a Black 99 x Fr(Lemma)30 b(15.)p Black 40 w FD(E[)p Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\025)f Fv(n)p FD(\(1)c Fs(\000)g Fv(\015)5 b FD(\))1823 1451 y Ft(n)p Fq(\000)p FC(1)p Black 523 1651 a Fp(Pr)l(o)l(of.)p Black 43 w FD(W)-7 b(e)37 b(ha)n(v)n(e)f(that)h(E[)p Fv(m)p FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])38 b(=)g(E[)1929 1589 y Fu(P)2016 1609 y Ft(n)2016 1676 y(i)p FC(=1)2142 1651 y Fv(I)2178 1663 y Ft(i)2206 1651 y FD(])h(=)2371 1589 y Fu(P)2458 1609 y Ft(n)2458 1676 y(i)p FC(=1)2584 1651 y FD(Prob)n([)14 b Fv(I)2833 1663 y Ft(i)2885 1651 y FD(=)22 b(1)14 b(])38 b Fs(\025)g Fv(n)p FD(\(1)24 b Fs(\000)523 1751 y Fv(\015)5 b FD(\))603 1721 y Ft(n)p Fq(\000)p FC(1)733 1751 y FD(,)28 b(b)n(y)f(Lemma)1195 1751 y SDict begin H.S end 1195 1751 a Black FD(14)p Black 1279 1697 a SDict begin H.R end 1279 1697 a 1279 1751 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.14) cvn H.B /ANN pdfmark end 1279 1751 a FD(.)2048 b Fs(u)-55 b(t)523 1842 y SDict begin H.S end 523 1842 a 523 1842 a SDict begin 12 H.A end 523 1842 a 523 1842 a SDict begin [/View [/XYZ H.V]/Dest (lemma.16) cvn /DEST pdfmark end 523 1842 a Black 78 x Fr(Lemma)30 b(16.)p Black 40 w Fp(F)-6 b(or)34 b(any)f Fv(\015)h Fs(\024)29 b Fv(t)p Fp(,)34 b Fv(t)29 b FD(=)g(0)p Fv(:)p FD(032704)p Fp(,)i(and)j(any)f Fv(n)c Fs(\025)h FD(2)2675 1849 y Fu(p)p 2757 1849 522 4 v 2757 1920 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)i FD(1)p Fv(=\015)37 b Fp(we)523 2020 y(have)31 b(that)1293 2080 y Fq(b)q Ft(n=)p FC(4)p Fq(c)1318 2112 y Fu(X)1324 2289 y Ft(i)p FC(=1)1476 2191 y Fv(p)1518 2203 y Ft(i)1569 2191 y Fs(\025)1666 2135 y Fv(n)p 1666 2172 50 4 v 1670 2248 a FD(4)1744 2191 y Fs(\001)1895 2135 y FD(1)p 1796 2172 241 4 v 1796 2248 a(1)18 b(+)g Fv(\015)5 b(n)2046 2074 y Fu(\022)2107 2191 y FD(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\))20 b Fs(\000)2518 2135 y FD(4)p 2514 2172 50 4 v 2514 2248 a Fv(n)2574 2074 y Fu(\023)523 2410 y Fp(wher)l(e)845 2655 y Fv(r)r FD(\()p Fv(\015)5 b FD(\))24 b(=)e Fv(k)1153 2513 y Fu( )1219 2504 y(s)p 1302 2504 1716 4 v 1312 2599 a FD(1)p 1312 2636 42 4 v 1312 2712 a(2)1377 2655 y(ln)1545 2599 y(1)p 1470 2636 191 4 v 1470 2712 a(1)c Fs(\000)g Fv(\015)1689 2655 y FD(+)1782 2599 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p 1782 2636 164 4 v 1843 2712 a(4)1956 2655 y(\()p Fv(\015)18 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\)\))18 b(+)2471 2599 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln)g(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2471 2636 489 4 v 2695 2712 a(4)2970 2655 y Fv(\015)3017 2513 y Fu(!)523 2916 y Fp(and)24 b Fv(k)i Fs(\031)d FD(2)p Fv(:)p FD(490795)p Fp(,)e(is)j(de\014ne)l(d)g(pr)l(e)l(cisely)h(in)f (the)g(pr)l(o)l(of)h(b)l(elow)g(and)f Fv(\014)t FD(\()p Fv(\015)5 b FD(\))24 b(=)f(1)6 b(+)3224 2883 y FC(2)p 3086 2897 310 4 v 3086 2944 a(ln)11 b(\(1)p Ft(=\015)t FC(\))o Fq(\000)p FC(1)523 3025 y Fp(is)30 b(de\014ne)g(in)g(L)l(emma) 1250 3041 y SDict begin H.S end 1250 3041 a Black -16 x Fp(27)p Black 1335 2971 a SDict begin H.R end 1335 2971 a 1335 3025 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.27) cvn H.B /ANN pdfmark end 1335 3025 a Fp(.)p Black 523 3231 a(Pr)l(o)l(of.)p Black 43 w FD(F)-7 b(rom)43 b(Lemma)1331 3231 y SDict begin H.S end 1331 3231 a Black FD(17)p Black 1414 3178 a SDict begin H.R end 1414 3178 a 1414 3231 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.17) cvn H.B /ANN pdfmark end 1414 3231 a 45 w FD(it)h(follo)n(ws)d (that)2045 3164 y Fu(\004)2094 3198 y Ft(n)p 2094 3212 42 4 v 2098 3260 a FC(4)2145 3164 y Fu(\005)2235 3231 y Fs(\024)2351 3169 y Fu(P)2439 3181 y Fs(b)2486 3153 y Fo(n)p 2486 3162 37 3 v 2490 3195 a Fb(4)2532 3181 y Fs(c)2439 3254 y Ft(i)p FC(=1)2587 3231 y Fv(p)2629 3243 y Ft(i)2657 3231 y FD(E[)o Fv(C)2795 3243 y Ft(i)2846 3231 y Fs(j)23 b Fv(I)2928 3243 y Ft(i)2979 3231 y FD(=)g(1].)44 b(Using)523 3331 y(Lemma)819 3331 y SDict begin H.S end 819 3331 a Black FD(18)p Black 902 3277 a SDict begin H.R end 902 3277 a 902 3331 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.18) cvn H.B /ANN pdfmark end 902 3331 a 28 w FD(to)28 b(upp)r(er)f(b)r(ound)h(E[)p Fv(C)1664 3343 y Ft(i)1715 3331 y Fs(j)23 b Fv(I)1797 3343 y Ft(i)1825 3331 y FD(])28 b(w)n(e)f(get)g(that:)896 3608 y SDict begin H.S end 896 3608 a 896 3608 a SDict begin 12 H.A end 896 3608 a 896 3608 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.71) cvn /DEST pdfmark end 896 3608 a -92 x Fu(j)949 3552 y Fv(n)p 949 3589 50 4 v 953 3665 a FD(4)1009 3516 y Fu(k)1076 3608 y Fs(\024)1164 3497 y Fq(b)p Ft(n=)p FC(4)q Fq(c)1189 3529 y Fu(X)1195 3706 y Ft(i)p FC(=1)1347 3608 y Fv(p)1389 3620 y Ft(i)1416 3441 y Fu(0)1416 3591 y(@)1503 3552 y FD(1)p 1499 3589 V 1499 3665 a Fv(n)1605 3608 y FD(+)45 b(1)18 b(+)2220 3552 y Fv(\015)5 b(n)p 1868 3589 802 4 v 1868 3697 a FD(1)18 b Fs(\000)2021 3665 y FC(4)p 2021 3679 34 4 v 2021 3726 a(3)2064 3606 y Fu(q)p 2147 3606 523 4 v 2157 3665 a FC(1)p 2157 3679 34 4 v 2157 3726 a(2)2214 3697 y FD(ln)2352 3665 y FC(1)p 2307 3679 124 4 v 2307 3726 a(1)p Fq(\000)p Ft(\015)2459 3697 y FD(+)2552 3665 y FC(ln)11 b Ft(n)p 2552 3679 108 4 v 2585 3726 a(n)2680 3441 y Fu(1)2680 3591 y(A)1076 3955 y Fs(\024)1178 3899 y FD(1)p 1174 3936 50 4 v 1174 4012 a Fv(n)1248 3844 y Fq(b)p Ft(n=)p FC(4)p Fq(c)1272 3876 y Fu(X)1316 4052 y FC(1)1430 3955 y Fv(p)1472 3967 y Ft(i)1546 3955 y FD(+)1657 3788 y Fu(0)1657 3938 y(@)1729 3955 y FD(1)18 b(+)2234 3899 y Fv(\015)5 b(n)p 1882 3936 802 4 v 1882 4044 a FD(1)18 b Fs(\000)2035 4012 y FC(4)p 2035 4026 34 4 v 2035 4073 a(3)2078 3953 y Fu(q)p 2161 3953 523 4 v 2171 4012 a FC(1)p 2171 4026 34 4 v 2171 4073 a(2)2228 4044 y FD(ln)2366 4012 y FC(1)p 2321 4026 124 4 v 2321 4073 a(1)p Fq(\000)p Ft(\015)2473 4044 y FD(+)2566 4012 y FC(ln)11 b Ft(n)p 2566 4026 108 4 v 2599 4073 a(n)2694 3788 y Fu(1)2694 3938 y(A)2780 3844 y Fq(b)q Ft(n=)p FC(4)p Fq(c)2805 3876 y Fu(X)2848 4052 y FC(1)2963 3955 y Fv(p)3005 3967 y Ft(i)1076 4295 y Fs(\024)1174 4239 y FD(1)p 1174 4276 42 4 v 1174 4352 a(4)1272 4295 y(+)1382 4178 y Fu(\022)1443 4295 y FD(1)18 b(+)1740 4239 y Fv(\015)5 b(n)p 1596 4276 385 4 v 1596 4352 a FD(1)18 b Fs(\000)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))1991 4178 y Fu(\023)2066 4184 y Fq(b)p Ft(n=)p FC(4)p Fq(c)2090 4216 y Fu(X)2134 4392 y FC(1)2249 4295 y Fv(p)2291 4307 y Ft(i)p Black 3257 4295 a FD(\(71\))p Black 523 4544 a(where)1404 4709 y Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))23 b(=)1766 4653 y(4)p 1766 4690 42 4 v 1766 4766 a(3)1818 4559 y Fu(s)p 1901 4559 624 4 v 1911 4653 a FD(1)p 1911 4690 42 4 v 1911 4766 a(2)1976 4709 y(ln)2144 4653 y(1)p 2069 4690 191 4 v 2069 4766 a(1)18 b Fs(\000)g Fv(\015)2288 4709 y FD(+)2381 4653 y(ln)13 b Fv(n)p 2381 4690 133 4 v 2423 4766 a(n)648 4923 y FD(No)n(w)27 b(w)n(e)g(giv)n(e)f(an)h(upp)r(er)h(b)r(ound)g(for) f(the)h(factor)e(1)18 b(+)2495 4886 y Ft(\015)t(n)p 2399 4904 271 4 v 2399 4951 a FC(1)p Fq(\000)p Ft(g)r FC(\()p Ft(n;\015)t FC(\))2708 4923 y FD(in)26 b(Equation)g(\()3195 4923 y SDict begin H.S end 3195 4923 a Black FD(71)p Black 3278 4869 a SDict begin H.R end 3278 4869 a 3278 4923 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.71) cvn H.B /ANN pdfmark end 3278 4923 a FD(\).)p Black 1922 5173 a(39)p Black eop end end %%Page: 40 40 TeXDict begin HPSdict begin 40 39 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.40) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(The)26 b(function)g Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))26 b(is)g(increasing)d(with)j Fv(\015)31 b FD(for)26 b Fv(\015)h Fs(2)d FD([0)p Fv(;)14 b(t)p FD(\))26 b(and)g(decreasing)e(with)i Fv(n)523 548 y FD(for)31 b Fv(n)e(>)g(e)i FD(\(use)h(Lemma)1377 548 y SDict begin H.S end 1377 548 a Black FD(25)p Black 1460 494 a SDict begin H.R end 1460 494 a 1460 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.25) cvn H.B /ANN pdfmark end 1460 548 a 31 w FD(with)f Fv(x)f FD(=)f(1)p Fv(=n)p FD(\).)i(Therefore)f(for)h Fv(n)e Fs(\025)g FD(2)2801 477 y Fu(p)p 2884 477 522 4 v 71 x Fv(e)14 b FD(ln)e(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)g FD(1)p Fv(=\015)523 702 y FD(and)31 b(for)g Fv(\015)j(<)c(t)h FD(w)n(e)g(ha)n(v)n(e)g(that) g Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))29 b Fs(\024)h Fv(g)1966 585 y Fu(\022)2026 702 y FD(2)2068 599 y Fu(q)p 2151 599 274 4 v 2161 662 a Ft(e)12 b FC(ln)e(\(1)p Ft(=t)p FC(\))p 2161 683 254 4 v 2275 731 a Ft(t)2424 702 y Fv(;)k(t)2491 585 y Fu(\023)2582 702 y Fs(\021)29 b Fv(h)g Fs(\031)h FD(0)p Fv(:)p FD(463713.)e(Since)523 874 y(for)c(an)n(y)g Fv(x)f Fs(6)p FD(=)g(1)h(w)n(e)g(ha)n(v)n(e)g(that)1564 841 y FC(1)p 1520 855 123 4 v 1520 902 a(1)p Fq(\000)p Ft(x)1675 874 y FD(=)f(1)12 b(+)g Fv(x)g FD(+)2067 841 y Ft(x)2105 816 y Fb(2)p 2041 855 V 2041 902 a FC(1)p Fq(\000)p Ft(x)2198 874 y FD(and)24 b(since)g Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))22 b Fs(6)p FD(=)h(1)h(w)n(e)g(get)h (that:)835 1138 y SDict begin H.S end 835 1138 a 835 1138 a SDict begin 12 H.A end 835 1138 a 835 1138 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.72) cvn /DEST pdfmark end 835 1138 a FD(1)18 b(+)1132 1082 y Fv(\015)5 b(n)p 988 1119 385 4 v 988 1195 a FD(1)18 b Fs(\000)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))1406 1138 y(=)23 b(1)18 b(+)g Fv(\015)5 b(n)1735 1021 y Fu(\022)1795 1138 y FD(1)18 b(+)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))18 b(+)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))2523 1104 y FC(2)2741 1082 y FD(1)p 2570 1119 V 2570 1195 a(1)18 b Fs(\000)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))2964 1021 y Fu(\023)1406 1370 y FD(=)23 b(\(1)18 b(+)g Fv(\015)5 b(n)p FD(\))18 b(+)g Fv(\015)5 b(n)18 b Fs(\001)h Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))2300 1253 y Fu(\022)2360 1370 y FD(1)18 b(+)2585 1314 y Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))p 2513 1351 V 2513 1427 a(1)18 b Fs(\000)g Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\))2908 1253 y Fu(\023)1406 1603 y Fs(\024)23 b FD(\(1)18 b(+)g Fv(\015)5 b(n)p FD(\))18 b(+)g Fv(\015)5 b(n)18 b Fs(\001)h Fv(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))2300 1486 y Fu(\022)2360 1603 y FD(1)18 b(+)2585 1547 y Fv(h)p 2513 1584 191 4 v 2513 1660 a FD(1)g Fs(\000)g Fv(h)2714 1486 y Fu(\023)1406 1835 y FD(=)23 b(\(1)18 b(+)g Fv(\015)5 b(n)p FD(\))1799 1718 y Fu(\022)1860 1835 y FD(1)18 b(+)2084 1779 y Fv(\015)5 b(n)p 2013 1816 241 4 v 2013 1892 a FD(1)18 b(+)g Fv(\015)5 b(n)2263 1835 y(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))2505 1718 y Fu(\022)2566 1835 y FD(1)18 b(+)2790 1779 y Fv(h)p 2719 1816 191 4 v 2719 1892 a FD(1)g Fs(\000)g Fv(h)2920 1718 y Fu(\023\023)3069 1835 y Fv(:)p Black 165 w FD(\(72\))p Black 648 2127 a(Next,)28 b(w)n(e)f(giv)n(e)e(an)j(upp)r(er)g(b)r(ound) g(on)1891 2034 y Fu(\020)1940 2127 y FD(1)18 b(+)2136 2094 y Ft(h)p 2093 2108 124 4 v 2093 2155 a FC(1)p Fq(\000)p Ft(h)2227 2034 y Fu(\021)2329 2090 y Ft(\015)t(n)p 2287 2108 164 4 v 2287 2155 a FC(1+)p Ft(\015)t(n)2460 2127 y Fv(g)s FD(\()p Fv(n;)c(\015)5 b FD(\).)975 2409 y SDict begin H.S end 975 2409 a 975 2409 a SDict begin 12 H.A end 975 2409 a 975 2409 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.73) cvn /DEST pdfmark end 975 2409 a -117 x Fu(\022)1036 2409 y FD(1)18 b(+)1260 2353 y Fv(h)p 1189 2390 191 4 v 1189 2466 a FD(1)g Fs(\000)g Fv(h)1390 2292 y Fu(\023)1532 2353 y Fv(\015)5 b(n)p 1461 2390 241 4 v 1461 2466 a FD(1)18 b(+)g Fv(\015)5 b(n)1711 2409 y(g)s FD(\()p Fv(n;)14 b(\015)5 b FD(\))1021 2677 y(=)1109 2560 y Fu(\022)1170 2677 y FD(1)18 b(+)1394 2621 y Fv(h)p 1323 2658 191 4 v 1323 2734 a FD(1)g Fs(\000)g Fv(h)1524 2560 y Fu(\023)1666 2621 y Fv(\015)5 b(n)p 1595 2658 241 4 v 1595 2734 a FD(1)18 b(+)g Fv(\015)5 b(n)1855 2621 y FD(4)p 1855 2658 42 4 v 1855 2734 a(3)1907 2528 y Fu(s)p 1990 2528 624 4 v 2000 2621 a FD(1)p 2000 2658 42 4 v 2000 2734 a(2)2065 2677 y(ln)2232 2621 y(1)p 2158 2658 191 4 v 2158 2734 a(1)18 b Fs(\000)g Fv(\015)2377 2677 y FD(+)2470 2621 y(ln)13 b Fv(n)p 2470 2658 133 4 v 2511 2734 a(n)1021 2972 y FD(=)23 b Fv(k)1155 2814 y Fu(s)p 1238 2814 1463 4 v 1248 2916 a FD(1)p 1248 2953 42 4 v 1248 3029 a(2)1313 2972 y(ln)1481 2916 y(1)p 1406 2953 191 4 v 1406 3029 a(1)18 b Fs(\000)g Fv(\015)1607 2855 y Fu(\022)1749 2916 y Fv(\015)5 b(n)p 1678 2953 241 4 v 1678 3029 a FD(1)18 b(+)g Fv(\015)5 b(n)1928 2855 y Fu(\023)1989 2872 y FC(2)2045 2972 y FD(+)2138 2916 y(ln)13 b Fv(n)p 2138 2953 133 4 v 2179 3029 a(n)2281 2855 y Fu(\022)2423 2916 y Fv(\015)5 b(n)p 2352 2953 241 4 v 2352 3029 a FD(1)18 b(+)g Fv(\015)5 b(n)2602 2855 y Fu(\023)2663 2872 y FC(2)1021 3257 y Fs(\024)23 b Fv(k)1155 3099 y Fu(s)p 1238 3099 1043 4 v 1248 3201 a FD(1)p 1248 3238 42 4 v 1248 3314 a(2)1313 3257 y(ln)1481 3201 y(1)p 1406 3238 191 4 v 1406 3314 a(1)18 b Fs(\000)g Fv(\015)1625 3257 y FD(+)1718 3201 y(ln)13 b Fv(n)p 1718 3238 133 4 v 1760 3314 a(n)1861 3140 y Fu(\022)2003 3201 y Fv(\015)5 b(n)p 1932 3238 241 4 v 1932 3314 a FD(1)18 b(+)g Fv(\015)5 b(n)2182 3140 y Fu(\023)2243 3158 y FC(2)1021 3532 y FD(=)23 b Fv(k)1155 3385 y Fu(s)p 1238 3385 874 4 v 1248 3476 a FD(1)p 1248 3513 42 4 v 1248 3589 a(2)1313 3532 y(ln)1481 3476 y(1)p 1406 3513 191 4 v 1406 3589 a(1)18 b Fs(\000)g Fv(\015)1625 3532 y FD(+)1812 3476 y Fv(n)c FD(ln)e Fv(n)p 1718 3513 384 4 v 1718 3589 a FD(\()p Fv(n)19 b FD(+)f(1)p Fv(=\015)5 b FD(\))2066 3565 y FC(2)1021 3822 y Fs(\024)23 b Fv(k)1155 3670 y Fu(s)p 1238 3670 1716 4 v 1248 3766 a FD(1)p 1248 3803 42 4 v 1248 3879 a(2)1313 3822 y(ln)1481 3766 y(1)p 1406 3803 191 4 v 1406 3879 a(1)18 b Fs(\000)g Fv(\015)1625 3822 y FD(+)1718 3766 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p 1718 3803 164 4 v 1779 3879 a(4)1891 3822 y(\()p Fv(\015)19 b FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))o(\))19 b(+)2407 3766 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln)g(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2407 3803 489 4 v 2631 3879 a(4)2906 3822 y Fv(\015)p Black 308 w FD(\(73\))p Black 1021 4012 a(=)23 b Fv(r)r FD(\()p Fv(\015)5 b FD(\))p Black 1997 w(\(74\))p Black 523 4236 a(where)34 b Fv(k)j Fs(\021)959 4204 y FC(4)p 959 4218 34 4 v 959 4265 a(3)1002 4144 y Fu(\020)1051 4236 y FD(1)18 b(+)1247 4204 y Ft(h)p 1204 4218 124 4 v 1204 4265 a FC(1)p Fq(\000)p Ft(h)1338 4144 y Fu(\021)1422 4236 y FD(and)34 b Fv(r)r FD(\()p Fv(\015)5 b FD(\))36 b(is)d(as)h(de\014ned)g (in)g(the)h(statemen)n(t)e(of)h(the)h(lemma)523 4361 y(ab)r(o)n(v)n(e.)26 b(Inequalit)n(y)f(\()1205 4361 y SDict begin H.S end 1205 4361 a Black FD(73)p Black 1289 4307 a SDict begin H.R end 1289 4307 a 1289 4361 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.73) cvn H.B /ANN pdfmark end 1289 4361 a FD(\))j(follo)n(ws)c(from)i(Lemma)2114 4361 y SDict begin H.S end 2114 4361 a Black FD(27)p Black 2197 4307 a SDict begin H.R end 2197 4307 a 2197 4361 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.27) cvn H.B /ANN pdfmark end 2197 4361 a FD(.)648 4469 y(Substituting)19 b(the)h(b)r(ound)h(from)d(Equation)h(\()2072 4469 y SDict begin H.S end 2072 4469 a Black FD(74)p Black 2155 4416 a SDict begin H.R end 2155 4416 a 2155 4469 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.73) cvn H.B /ANN pdfmark end 2155 4469 a FD(\))h(in)n(to)f(Equation)f(\()2751 4469 y SDict begin H.S end 2751 4469 a Black FD(72)p Black 2835 4416 a SDict begin H.R end 2835 4416 a 2835 4469 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.72) cvn H.B /ANN pdfmark end 2835 4469 a FD(\))i(and)g(the)h(b)r(ound)523 4569 y(from)26 b(Equation)g(\()1110 4569 y SDict begin H.S end 1110 4569 a Black FD(72)p Black 1193 4515 a SDict begin H.R end 1193 4515 a 1193 4569 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.72) cvn H.B /ANN pdfmark end 1193 4569 a FD(\))i(in)n(to)f(Equation)e(\()1812 4569 y SDict begin H.S end 1812 4569 a Black FD(71)p Black 1895 4515 a SDict begin H.R end 1895 4515 a 1895 4569 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.71) cvn H.B /ANN pdfmark end 1895 4569 a FD(\))j(w)n(e)g(get)1283 4768 y Fu(j)1337 4804 y Fv(n)p 1337 4842 50 4 v 1341 4918 a FD(4)1397 4768 y Fu(k)1464 4861 y Fs(\024)1561 4804 y FD(1)p 1561 4842 42 4 v 1561 4918 a(4)1631 4861 y(+)18 b(\(1)g(+)g Fv(\015)5 b(n)p FD(\)\(1)18 b(+)g Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))2391 4750 y Fq(b)r Ft(n=)p FC(4)p Fq(c)2417 4782 y Fu(X)2460 4958 y FC(1)2575 4861 y Fv(p)2617 4873 y Ft(i)p Black 1922 5173 a FD(40)p Black eop end end %%Page: 41 41 TeXDict begin HPSdict begin 41 40 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.41) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Isolating)987 386 y Fu(P)1089 448 y Fv(p)1131 460 y Ft(i)1186 448 y FD(w)n(e)27 b(obtain)1207 717 y SDict begin H.S end 1207 717 a 1207 717 a SDict begin 12 H.A end 1207 717 a 1207 717 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.75) cvn /DEST pdfmark end 1207 717 a 1221 606 a Fq(b)p Ft(n=)p FC(4)p Fq(c)1245 638 y Fu(X)1288 814 y FC(1)1403 717 y Fv(p)1445 729 y Ft(i)1496 717 y Fs(\025)1584 599 y Fu(\022)1645 624 y(j)1699 660 y Fv(n)p 1699 697 50 4 v 1703 774 a FD(4)1758 624 y Fu(k)1821 717 y Fs(\000)1914 660 y FD(1)p 1914 697 42 4 v 1914 774 a(4)1965 599 y Fu(\023)2348 660 y FD(1)p 2036 697 665 4 v 2036 774 a(\(1)19 b(+)f Fv(\015)5 b(n)p FD(\)\(1)17 b(+)i Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))1496 975 y Fs(\025)1584 858 y Fu(\022)1645 883 y(j)1699 919 y Fv(n)p 1699 956 50 4 v 1703 1032 a FD(4)1758 883 y Fu(k)1821 975 y Fs(\000)1914 919 y FD(1)p 1914 956 42 4 v 1914 1032 a(4)1965 858 y Fu(\023)2168 919 y FD(1)p 2036 956 305 4 v 2036 1032 a(\(1)19 b(+)f Fv(\015)5 b(n)p FD(\))2351 975 y(\(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))p Black 548 w(\(75\))p Black 1496 1208 a Fs(\025)1584 1091 y Fu(\022)1655 1152 y Fv(n)p 1655 1189 50 4 v 1659 1265 a FD(4)1733 1208 y Fs(\000)1826 1152 y FD(3)p 1826 1189 42 4 v 1826 1265 a(4)1896 1208 y Fs(\000)1989 1152 y FD(1)p 1989 1189 V 1989 1265 a(4)2040 1091 y Fu(\023)2211 1152 y FD(1)p 2112 1189 241 4 v 2112 1265 a(1)17 b(+)i Fv(\015)5 b(n)2362 1208 y FD(\(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))1496 1430 y(=)1584 1337 y Fu(\020)1643 1373 y Fv(n)p 1643 1411 50 4 v 1647 1487 a FD(4)1722 1430 y Fs(\000)18 b FD(1)1847 1337 y Fu(\021)2005 1373 y FD(1)p 1906 1411 241 4 v 1906 1487 a(1)g(+)g Fv(\015)5 b(n)2156 1430 y FD(\(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))1496 1646 y Fs(\025)1594 1589 y Fv(n)p 1594 1626 50 4 v 1598 1703 a FD(4)1654 1646 y(\(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\)\))19 b Fs(\000)2224 1589 y FD(1)p 2124 1626 241 4 v 2124 1703 a(1)f(+)g Fv(\015)5 b(n)1496 1872 y FD(=)1594 1816 y Fv(n)p 1594 1853 50 4 v 1598 1929 a FD(4)1672 1872 y Fs(\001)1823 1816 y FD(1)p 1723 1853 241 4 v 1723 1929 a(1)18 b(+)g Fv(\015)5 b(n)1974 1755 y Fu(\022)2035 1872 y FD(1)18 b Fs(\000)g Fv(r)r FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)2445 1816 y FD(4)p 2441 1853 50 4 v 2441 1929 a Fv(n)2501 1755 y Fu(\023)523 2098 y FD(where)27 b(Inequalit)n(y)e(\()1187 2098 y SDict begin H.S end 1187 2098 a Black FD(75)p Black 1270 2045 a SDict begin H.R end 1270 2045 a 1270 2098 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.75) cvn H.B /ANN pdfmark end 1270 2098 a FD(\))j(follo)n(ws)d(since)h(1)p Fv(=)p FD(\(1)18 b(+)g Fv(x)p FD(\))24 b Fs(\024)e FD(1)c Fs(\000)g Fv(x)28 b FD(for)f Fv(x)d Fs(\025)e(\000)p FD(1.)462 b Fs(u)-55 b(t)523 2187 y SDict begin H.S end 523 2187 a 523 2187 a SDict begin 12 H.A end 523 2187 a 523 2187 a SDict begin [/View [/XYZ H.V]/Dest (lemma.17) cvn /DEST pdfmark end 523 2187 a Black 90 x Fr(Lemma)30 b(17.)p Black 40 w Fp(F)-6 b(or)30 b(al)t(l)h FD(1)22 b Fs(\024)h Fv(k)j Fs(\024)d Fv(n)p Fp(,)1713 2215 y Fu(P)1800 2236 y Ft(k)1800 2302 y(i)p FC(=1)1926 2277 y Fv(p)1968 2289 y Ft(i)1995 2277 y FD(E[)p Fv(C)2134 2289 y Ft(i)2185 2277 y Fs(j)g Fv(I)2267 2289 y Ft(i)2318 2277 y FD(=)g(1])f Fs(\025)h Fv(k)s Fp(.)p Black 523 2445 a(Pr)l(o)l(of.)p Black 43 w FD(Recall)h(that)j Fv(X)1280 2457 y Ft(j)1342 2445 y FD(is)e(a)h(random)f(v)-5 b(ariable)24 b(equal)h(to)h(the)h Fv(j)5 b FD(th)27 b(p)r(oin)n(t)f(from)f(the)i (left.)523 2545 y(F)-7 b(or)25 b(a)h(giv)n(en)e(instan)n(tiation)e(of)k (the)g Fv(X)1741 2557 y Ft(j)1802 2545 y FD(let)f Fv(Q)1986 2557 y Ft(k)2053 2545 y FD(b)r(e)h(the)g(subset)g(of)g(the)g Fv(k)j FD(leftmost)24 b(p)r(oin)n(ts)523 2644 y(that)i(w)n(ere)e(c)n (hosen)h(b)n(y)g(the)g(greedy)g(algorithm.)c(Note)k(that)h Fv(i)d Fs(2)g Fv(Q)2642 2656 y Ft(k)2708 2644 y FD(i\013)i(the)h (instan)n(tiation)523 2744 y(of)i Fv(I)654 2756 y Ft(i)709 2744 y FD(is)f(one.)648 2843 y(The)34 b(union)g(of)h(the)g(in)n(terv)-5 b(als)32 b([)p Fv(X)1747 2855 y Ft(j)1782 2843 y Fv(;)14 b(X)1888 2855 y Ft(j)1945 2843 y FD(+)23 b Fv(\015)5 b FD(\),)35 b Fv(j)40 b Fs(2)35 b Fv(Q)2401 2855 y Ft(k)2442 2843 y FD(,)f(m)n(ust)g(include)f(the)i(instan-)523 2943 y(tiations)29 b(of)h Fv(X)993 2955 y FC(1)1030 2943 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)1284 2955 y Ft(k)1325 2943 y FD(.)31 b(This)e(follo)n(ws)f(since)i(otherwise)f(there)i(w)n(ould)e (b)r(e)j(some)d(p)r(oin)n(t)523 3043 y(that)j(w)n(as)e(not)i(selected)e (b)n(y)h(the)h(greedy)e(c)n(hoice)g(and)h(that)h(could)e(b)r(e)i (selected,)f(con)n(tra-)523 3142 y(dicting)j(the)i(de\014nition)e(of)h (the)h(greedy)f(algorithm.)c(Hence:)2550 3080 y Fu(P)2637 3101 y Ft(k)2637 3167 y(i)p FC(=1)2763 3142 y Fv(C)2822 3154 y Ft(i)2850 3142 y Fv(I)2886 3154 y Ft(i)2950 3142 y Fs(\025)36 b Fv(k)s FD(.)g(T)-7 b(aking)523 3242 y(Exp)r(ectations)26 b(w)n(e)h(get)g(that)665 3495 y Fv(k)f Fs(\024)863 3391 y Ft(k)821 3416 y Fu(X)828 3593 y Ft(i)p FC(=1)955 3495 y FD(E[)p Fv(C)1094 3507 y Ft(i)1140 3495 y Fs(\001)19 b Fv(I)1218 3507 y Ft(i)1246 3495 y FD(])734 3787 y(=)863 3683 y Ft(k)821 3708 y Fu(X)828 3885 y Ft(i)p FC(=1)955 3787 y FD(Prob)o([)14 b Fv(I)1205 3799 y Ft(i)1256 3787 y FD(=)23 b(0)14 b(])j Fs(\001)i FD(E[)p Fv(C)1621 3799 y Ft(i)1667 3787 y Fs(\001)f FD(0)23 b Fs(j)g Fv(I)1855 3799 y Ft(i)1906 3787 y FD(=)g(0])18 b(+)g(Prob)o([)c Fv(I)2410 3799 y Ft(i)2461 3787 y FD(=)22 b(1)14 b(])k Fs(\001)h FD(E[)o Fv(C)2825 3799 y Ft(i)2872 3787 y Fs(\001)f FD(1)23 b Fs(j)g Fv(I)3060 3799 y Ft(i)3111 3787 y FD(=)g(1])734 4079 y(=)863 3975 y Ft(k)821 4000 y Fu(X)828 4177 y Ft(i)p FC(=1)955 4079 y Fv(p)997 4091 y Ft(i)1025 4079 y FD(E[)o Fv(C)1163 4091 y Ft(i)1214 4079 y Fs(j)g Fv(I)1296 4091 y Ft(i)1348 4079 y FD(=)f(1])p Fv(:)3350 4316 y Fs(u)-55 b(t)523 4384 y SDict begin H.S end 523 4384 a 523 4384 a SDict begin 12 H.A end 523 4384 a 523 4384 a SDict begin [/View [/XYZ H.V]/Dest (lemma.18) cvn /DEST pdfmark end 523 4384 a Black 100 x Fr(Lemma)30 b(18.)p Black 40 w Fp(Given)i(any)h FD(0)26 b Fv(<)g(\015)31 b(<)26 b FD(0)p Fv(:)p FD(032704)p Fp(,)j(inte)l(ger)j Fv(n)26 b(>)g FD(2)2612 4413 y Fu(p)p 2695 4413 522 4 v 71 x Fv(e)14 b FD(ln)e(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\001)h FD(1)p Fv(=\015)s Fp(,)32 b(and)523 4583 y(inte)l(ger)e FD(0)22 b Fs(\024)h Fv(i)g Fs(\024)f Fv(n=)p FD(4)29 b Fp(we)h(have)h(that)1060 4781 y FD(E[)p Fv(C)1199 4793 y Ft(i)1250 4781 y Fs(j)23 b Fv(I)1332 4793 y Ft(i)1383 4781 y FD(=)g(1])f Fs(\024)h FD(1)18 b(+)2151 4725 y Fv(\015)5 b(n)p 1799 4762 802 4 v 1799 4870 a FD(1)18 b Fs(\000)1952 4837 y FC(4)p 1952 4851 34 4 v 1952 4899 a(3)1995 4778 y Fu(q)p 2078 4778 523 4 v 2088 4837 a FC(1)p 2088 4851 34 4 v 2088 4899 a(2)2145 4870 y FD(ln)2283 4837 y FC(1)p 2238 4851 124 4 v 2238 4899 a(1)p Fq(\000)p Ft(\015)2390 4870 y FD(+)2483 4837 y FC(ln)10 b Ft(n)p 2483 4851 108 4 v 2516 4899 a(n)2629 4781 y FD(+)18 b(1)p Fv(=n:)p Black 1922 5173 a FD(41)p Black eop end end %%Page: 42 42 TeXDict begin HPSdict begin 42 41 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.42) cvn /DEST pdfmark end 515 125 a Black Black Black 523 448 a Fp(Pr)l(o)l(of.)p Black 43 w FD(De\014ne)1362 691 y SDict begin H.S end 1362 691 a 1362 691 a SDict begin 12 H.A end 1362 691 a 1362 691 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.76) cvn /DEST pdfmark end 1362 691 a Fv(a)1406 703 y Ft(i)1456 691 y FD(=)1554 635 y Fv(i)18 b Fs(\000)g FD(1)p 1554 672 172 4 v 1615 748 a Fv(n)1754 691 y FD(+)1837 542 y Fu(s)p 1920 542 624 4 v 1930 635 a FD(1)p 1930 672 42 4 v 1930 748 a(2)1995 691 y(ln)2163 635 y(1)p 2088 672 191 4 v 2088 748 a(1)g Fs(\000)g Fv(\015)2307 691 y FD(+)2400 635 y(ln)13 b Fv(n)p 2400 672 133 4 v 2442 748 a(n)2543 691 y(:)691 b FD(\(76\))523 966 y(W)-7 b(e)28 b(ha)n(v)n(e)e(that)651 1190 y SDict begin H.S end 651 1190 a 651 1190 a SDict begin 12 H.A end 651 1190 a 651 1190 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.77) cvn /DEST pdfmark end 651 1190 a FD(E[)p Fv(C)790 1202 y Ft(i)841 1190 y Fs(j)d Fv(I)923 1202 y Ft(i)974 1190 y FD(=)g(1])g(=)106 b(Prob)n([)14 b(\()p Fv(X)1635 1202 y Ft(i)1686 1190 y Fv(<)23 b(a)1818 1202 y Ft(i)1845 1190 y FD(\))h Fs(j)f Fv(I)1983 1202 y Ft(i)2034 1190 y FD(=)f(1)14 b(])k Fs(\001)h FD(E[)o Fv(C)2398 1202 y Ft(i)2449 1190 y Fs(j)24 b FD(\()p Fv(X)2597 1202 y Ft(i)2647 1190 y Fv(<)f(a)2779 1202 y Ft(i)2806 1190 y FD(\))c Fs(^)g Fv(I)2967 1202 y Ft(i)3018 1190 y FD(=)k(1])1404 1315 y(+Prob)n([)14 b Fv(X)1751 1327 y Ft(i)1801 1315 y Fs(\025)23 b Fv(a)1933 1327 y Ft(i)1984 1315 y Fs(j)g Fv(I)2066 1327 y Ft(i)2117 1315 y FD(=)f(1)14 b(])k Fs(\001)h FD(E[)o Fv(C)2481 1327 y Ft(i)2532 1315 y Fs(j)24 b FD(\()p Fv(X)2680 1327 y Ft(i)2730 1315 y Fs(\025)f Fv(a)2862 1327 y Ft(i)2890 1315 y FD(\))18 b Fs(^)h Fv(I)3050 1327 y Ft(i)3101 1315 y FD(=)k(1])o Fv(:)p Black -19 w FD(\(77\))p Black 523 1539 a(F)-7 b(or)29 b(the)h(second)f(term,)g(from)g(Lemma) 1810 1539 y SDict begin H.S end 1810 1539 a Black FD(19)p Black 1893 1486 a SDict begin H.R end 1893 1486 a 1893 1539 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.19) cvn H.B /ANN pdfmark end 1893 1539 a FD(,)h(w)n(e)f(deriv)n(e)f(that)i (Prob)o([)14 b Fv(X)2782 1551 y Ft(i)2832 1539 y Fs(\025)23 b Fv(a)2964 1551 y Ft(i)3014 1539 y Fs(j)g Fv(I)3096 1551 y Ft(i)3148 1539 y FD(=)f(1)14 b(])26 b Fs(\024)553 1606 y FC(1)p 533 1620 74 4 v 533 1667 a Ft(n)574 1651 y Fb(2)616 1639 y FD(.)j(In)g(addition,)e(since)g(the)i(n)n(um)n(b)r (er)e(of)i(p)r(oin)n(ts)e(to)i(the)g(righ)n(t)e(of)h(\(or)g(at\))h(the) g Fv(i)p FD(th)g(p)r(oin)n(t)523 1738 y(from)d(the)i(left)g(is)e Fv(n)18 b Fs(\000)g Fv(i)g FD(+)g(1,)28 b(w)n(e)f(ha)n(v)n(e)f(that) 1200 1963 y(E[)p Fv(C)1339 1975 y Ft(i)1390 1963 y Fs(j)d FD(\()p Fv(X)1537 1975 y Ft(i)1588 1963 y Fs(\025)f Fv(a)1719 1975 y Ft(i)1747 1963 y FD(\))c Fs(^)h Fv(I)1907 1975 y Ft(i)1958 1963 y FD(=)k(1])g Fs(\024)f Fv(n)d Fs(\000)f Fv(i)g FD(+)g(1)k Fs(\024)h Fv(n:)523 2187 y FD(Hence,)28 b(for)f(the)h(second)f(term)970 2446 y SDict begin H.S end 970 2446 a 970 2446 a SDict begin 12 H.A end 970 2446 a 970 2446 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.78) cvn /DEST pdfmark end 970 2446 a FD(Prob)n([)14 b Fv(X)1252 2458 y Ft(i)1303 2446 y Fs(\025)22 b Fv(a)1434 2458 y Ft(i)1485 2446 y Fs(j)h Fv(I)1567 2458 y Ft(i)1618 2446 y FD(=)g(1)14 b(])k Fs(\001)g FD(E[)p Fv(C)1983 2458 y Ft(i)2034 2446 y Fs(j)23 b FD(\()p Fv(X)2181 2458 y Ft(i)2232 2446 y Fs(\025)f Fv(a)2363 2458 y Ft(i)2391 2446 y FD(\))d Fs(^)f Fv(I)2551 2458 y Ft(i)2603 2446 y FD(=)k(1])h Fs(\024)2880 2390 y FD(1)p 2875 2427 50 4 v 2875 2503 a Fv(n)2935 2446 y(:)299 b FD(\(78\))648 2719 y(F)-7 b(or)31 b(the)i(\014rst)f(term)f(w)n(e)g(apply)g(Lemmata) 2062 2719 y SDict begin H.S end 2062 2719 a Black FD(21)p Black 2145 2665 a SDict begin H.R end 2145 2665 a 2145 2719 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.21) cvn H.B /ANN pdfmark end 2145 2719 a 32 w FD(\(in)h(Equation)e(\()2706 2719 y SDict begin H.S end 2706 2719 a Black FD(79)p Black 2789 2665 a SDict begin H.R end 2789 2665 a 2789 2719 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.79) cvn H.B /ANN pdfmark end 2789 2719 a FD(\)\))j(and)3052 2719 y SDict begin H.S end 3052 2719 a Black FD(22)p Black 3135 2665 a SDict begin H.R end 3135 2665 a 3135 2719 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.22) cvn H.B /ANN pdfmark end 3135 2719 a 32 w FD(\(in)f(In-)523 2818 y(equalit)n(y)25 b(\()871 2818 y SDict begin H.S end 871 2818 a Black FD(80)p Black 954 2765 a SDict begin H.R end 954 2765 a 954 2818 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.79) cvn H.B /ANN pdfmark end 954 2818 a FD(\)\))k(to)e(deriv) n(e)790 3043 y SDict begin H.S end 790 3043 a 790 3043 a SDict begin 12 H.A end 790 3043 a 790 3043 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.79) cvn /DEST pdfmark end 790 3043 a FD(E[)o Fv(C)928 3055 y Ft(i)979 3043 y Fs(j)c FD(\()p Fv(X)1126 3055 y Ft(i)1177 3043 y Fv(<)g(a)1309 3055 y Ft(i)1336 3043 y FD(\))c Fs(^)g Fv(I)1497 3055 y Ft(i)1548 3043 y FD(=)k(1])813 3214 y(=)901 3101 y Fu(Z)984 3122 y Ft(a)1020 3130 y Fo(i)947 3290 y FC(0)1064 3214 y FD(Prob)o([)14 b Fv(X)1347 3226 y Ft(i)1397 3214 y FD(=)23 b Fv(x)g Fs(j)g Fv(X)1670 3226 y Ft(i)1721 3214 y Fv(<)g(a)1853 3226 y Ft(i)1899 3214 y Fs(^)18 b Fv(I)2008 3226 y Ft(i)2060 3214 y FD(=)k(1)14 b(]E[)o Fv(C)2364 3226 y Ft(i)2415 3214 y Fs(j)23 b Fv(X)2530 3226 y Ft(i)2581 3214 y FD(=)f Fv(x)d Fs(^)g Fv(I)2844 3226 y Ft(i)2895 3214 y FD(=)k(1])p Fv(dx)813 3441 y FD(=)901 3328 y Fu(Z)984 3348 y Ft(a)1020 3356 y Fo(i)947 3516 y FC(0)1064 3441 y FD(Prob)o([)14 b Fv(X)1347 3453 y Ft(i)1397 3441 y FD(=)23 b Fv(x)g Fs(j)g Fv(X)1670 3453 y Ft(i)1721 3441 y Fv(<)g(a)1853 3453 y Ft(i)1899 3441 y Fs(^)18 b Fv(I)2008 3453 y Ft(i)2060 3441 y FD(=)k(1)14 b(]E[)o Fv(C)2364 3453 y Ft(i)2415 3441 y Fs(j)23 b Fv(X)2530 3453 y Ft(i)2581 3441 y FD(=)f Fv(x)p FD(])q Fv(dx)p Black 428 w FD(\(79\))p Black 813 3670 a Fs(\024)901 3557 y Fu(Z)984 3577 y Ft(a)1020 3585 y Fo(i)947 3746 y FC(0)1064 3670 y FD(Prob)o([)14 b Fv(X)1347 3682 y Ft(i)1397 3670 y FD(=)23 b Fv(x)g Fs(j)g Fv(X)1670 3682 y Ft(i)1721 3670 y Fv(<)g(a)1853 3682 y Ft(i)1899 3670 y Fs(^)18 b Fv(I)2008 3682 y Ft(i)2060 3670 y FD(=)k(1)14 b(])2226 3553 y Fu(\022)2287 3670 y FD(1)k(+)2502 3614 y Fv(\015)p 2440 3651 173 4 v 2440 3727 a FD(1)g Fs(\000)g Fv(t)2623 3670 y FD(\()p Fv(n)g Fs(\000)g Fv(i)p FD(\))2867 3553 y Fu(\023)2928 3670 y Fv(dx)p Black 239 w FD(\(80\))p Black 813 3902 a Fs(\024)901 3789 y Fu(Z)984 3810 y Ft(a)1020 3818 y Fo(i)947 3978 y FC(0)1064 3902 y FD(Prob)o([)c Fv(X)1347 3914 y Ft(i)1397 3902 y FD(=)23 b Fv(x)g Fs(j)g Fv(X)1670 3914 y Ft(i)1721 3902 y Fv(<)g(a)1853 3914 y Ft(i)1899 3902 y Fs(^)18 b Fv(I)2008 3914 y Ft(i)2060 3902 y FD(=)k(1)14 b(])2226 3785 y Fu(\022)2287 3902 y FD(1)k(+)2523 3846 y Fv(\015)p 2440 3883 215 4 v 2440 3959 a FD(1)g Fs(\000)g Fv(a)2627 3971 y Ft(i)2664 3902 y FD(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))2909 3785 y Fu(\023)2970 3902 y Fv(dx)813 4135 y Fs(\024)901 4018 y Fu(\022)962 4135 y FD(1)g(+)1198 4079 y Fv(\015)p 1115 4116 V 1115 4192 a FD(1)g Fs(\000)g Fv(a)1302 4204 y Ft(i)1339 4135 y FD(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))1584 4018 y Fu(\023)1659 4022 y(Z)1742 4042 y Ft(a)1778 4050 y Fo(i)1705 4210 y FC(0)1822 4135 y FD(Prob)o([)c Fv(X)2105 4147 y Ft(i)2155 4135 y FD(=)23 b Fv(x)g Fs(j)g Fv(X)2428 4147 y Ft(i)2479 4135 y Fv(<)g(a)2611 4147 y Ft(i)2657 4135 y Fs(^)18 b Fv(I)2766 4147 y Ft(i)2818 4135 y FD(=)k(1)14 b(])p Fv(dx)813 4339 y FD(=)23 b(1)18 b(+)1137 4283 y Fv(\015)p 1054 4320 V 1054 4396 a FD(1)g Fs(\000)g Fv(a)1241 4408 y Ft(i)1278 4339 y FD(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))p Black 1734 w(\(81\))p Black 648 4624 a(By)27 b(substituting)f(\()1272 4624 y SDict begin H.S end 1272 4624 a Black FD(78)p Black 1355 4571 a SDict begin H.R end 1355 4571 a 1355 4624 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.78) cvn H.B /ANN pdfmark end 1355 4624 a FD(\))i(and)g(\()1609 4624 y SDict begin H.S end 1609 4624 a Black FD(81)p Black 1692 4571 a SDict begin H.R end 1692 4571 a 1692 4624 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.79) cvn H.B /ANN pdfmark end 1692 4624 a FD(\))g(in)n(to)e(Equation)g(\()2311 4624 y SDict begin H.S end 2311 4624 a Black FD(77)p Black 2394 4571 a SDict begin H.R end 2394 4571 a 2394 4624 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.77) cvn H.B /ANN pdfmark end 2394 4624 a FD(\))i(w)n(e)f(get)h(that:)1319 4896 y SDict begin H.S end 1319 4896 a 1319 4896 a SDict begin 12 H.A end 1319 4896 a 1319 4896 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.82) cvn /DEST pdfmark end 1319 4896 a FD(E[)o Fv(C)1457 4908 y Ft(i)1508 4896 y Fs(j)23 b Fv(I)1590 4908 y Ft(i)1642 4896 y FD(=)f(1])h Fs(\024)g FD(1)18 b(+)2058 4840 y Fv(\015)5 b FD(\()p Fv(n)18 b Fs(\000)g Fv(i)p FD(\))p 2058 4877 293 4 v 2096 4953 a(1)g Fs(\000)g Fv(a)2283 4965 y Ft(i)2406 4896 y FD(+)2531 4840 y(1)p 2526 4877 50 4 v 2526 4953 a Fv(n)2586 4896 y(:)648 b FD(\(82\))p Black 1922 5173 a(42)p Black eop end end %%Page: 43 43 TeXDict begin HPSdict begin 43 42 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.43) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Substituting)26 b(the)i(v)-5 b(alue)27 b(of)g Fv(a)1619 460 y Ft(i)1674 448 y FD(from)g(Equation)e (\()2261 448 y SDict begin H.S end 2261 448 a Black FD(76)p Black 2344 395 a SDict begin H.R end 2344 395 a 2344 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.76) cvn H.B /ANN pdfmark end 2344 448 a FD(\))j(in)f(Equation)f(\()2892 448 y SDict begin H.S end 2892 448 a Black FD(82)p Black 2975 395 a SDict begin H.R end 2975 395 a 2975 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.82) cvn H.B /ANN pdfmark end 2975 448 a FD(\))i(w)n(e)f(get)1170 734 y SDict begin H.S end 1170 734 a 1170 734 a SDict begin 12 H.A end 1170 734 a 1170 734 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.83) cvn /DEST pdfmark end 1170 734 a 1180 678 a Fv(\015)5 b FD(\()p Fv(n)18 b Fs(\000)h Fv(i)p FD(\))p 1180 715 293 4 v 1219 791 a(1)f Fs(\000)g Fv(a)1406 803 y Ft(i)1506 734 y FD(=)1947 678 y Fv(\015)5 b FD(\()p Fv(n)18 b Fs(\000)g Fv(i)p FD(\))p 1603 715 979 4 v 1603 823 a(1)g Fs(\000)1756 790 y Ft(i)p Fq(\000)p FC(1)p 1756 804 109 4 v 1790 852 a Ft(n)1893 823 y Fs(\000)1976 731 y Fu(q)p 2059 731 523 4 v 2069 790 a FC(1)p 2069 804 34 4 v 2069 852 a(2)2126 823 y FD(ln)2264 790 y FC(1)p 2219 804 124 4 v 2219 852 a(1)p Fq(\000)p Ft(\015)2371 823 y FD(+)2464 790 y FC(ln)11 b Ft(n)p 2464 804 108 4 v 2497 852 a(n)1506 1029 y FD(=)2005 973 y Fv(\015)5 b(n)p FD(\()p Fv(n)18 b Fs(\000)g Fv(i)p FD(\))p 1603 1010 1145 4 v 1603 1118 a(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))g(+)g(1)g Fs(\000)g Fv(n)2142 1027 y Fu(q)p 2225 1027 523 4 v 2235 1086 a FC(1)p 2235 1100 34 4 v 2235 1147 a(2)2292 1118 y FD(ln)2430 1086 y FC(1)p 2385 1100 124 4 v 2385 1147 a(1)p Fq(\000)p Ft(\015)2537 1118 y FD(+)2630 1086 y FC(ln)11 b Ft(n)p 2630 1100 108 4 v 2663 1147 a(n)1506 1324 y Fs(\024)1933 1268 y Fv(\015)5 b(n)p FD(\()p Fv(n)18 b Fs(\000)g Fv(i)p FD(\))p 1603 1305 1002 4 v 1603 1414 a(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))g Fs(\000)g Fv(n)1999 1322 y Fu(q)p 2082 1322 523 4 v 2092 1381 a FC(1)p 2092 1395 34 4 v 2092 1442 a(2)2149 1414 y FD(ln)2287 1381 y FC(1)p 2242 1395 124 4 v 2242 1442 a(1)p Fq(\000)p Ft(\015)2394 1414 y FD(+)2487 1381 y FC(ln)11 b Ft(n)p 2487 1395 108 4 v 2520 1442 a(n)1506 1593 y FD(=)1899 1537 y Fv(\015)5 b(n)p 1603 1574 689 4 v 1603 1717 a FD(1)18 b Fs(\000)1756 1654 y Ft(n)1797 1590 y Fa(q)p 1863 1590 419 3 v 1873 1632 a Fb(1)p 1873 1641 29 3 v 1873 1674 a(2)1922 1654 y FC(ln)2038 1632 y Fb(1)p 1999 1641 108 3 v 1999 1674 a(1)p Fd(\000)p Fo(\015)2116 1654 y FC(+)2177 1632 y Fb(ln)10 b Fo(n)p 2177 1641 95 3 v 2206 1674 a(n)p 1756 1698 526 4 v 1961 1746 a Ft(n)p Fq(\000)p Ft(i)1506 1878 y FD(=)1899 1822 y Fv(\015)5 b(n)p 1603 1859 689 4 v 1603 2002 a FD(1)18 b Fs(\000)1756 1939 y Ft(n)1797 1874 y Fa(q)p 1863 1874 419 3 v 1873 1916 a Fb(1)p 1873 1925 29 3 v 1873 1959 a(2)1922 1939 y FC(ln)2038 1916 y Fb(1)p 1999 1925 108 3 v 1999 1959 a(1)p Fd(\000)p Fo(\015)2116 1939 y FC(+)2177 1916 y Fb(ln)10 b Fo(n)p 2177 1925 95 3 v 2206 1959 a(n)p 1756 1983 526 4 v 1961 2031 a Ft(n)p Fq(\000)p Ft(i)2301 1878 y Fv(:)p Black 933 w FD(\(83\))p Black 648 2274 a(Since)26 b Fv(i)d Fs(\024)g Fv(n=)p FD(4)j(w)n(e)h(get)h(that:)1176 2626 y SDict begin H.S end 1176 2626 a 1176 2626 a SDict begin 12 H.A end 1176 2626 a 1176 2626 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.84) cvn /DEST pdfmark end 1176 2626 a 1186 2539 a Fv(n)1236 2448 y Fu(q)p 1318 2448 523 4 v 1328 2507 a FC(1)p 1328 2521 34 4 v 1328 2568 a(2)1385 2539 y FD(ln)1523 2507 y FC(1)p 1478 2521 124 4 v 1478 2568 a(1)p Fq(\000)p Ft(\015)1630 2539 y FD(+)1723 2507 y FC(ln)11 b Ft(n)p 1723 2521 108 4 v 1756 2568 a(n)p 1186 2607 656 4 v 1423 2683 a Fv(n)18 b Fs(\000)h Fv(i)1874 2626 y Fs(\024)1972 2539 y Fv(n)2022 2448 y Fu(q)p 2105 2448 523 4 v 2115 2507 a FC(1)p 2115 2521 34 4 v 2115 2568 a(2)2172 2539 y FD(ln)2310 2507 y FC(1)p 2265 2521 124 4 v 2265 2568 a(1)p Fq(\000)p Ft(\015)2416 2539 y FD(+)2509 2507 y FC(ln)11 b Ft(n)p 2509 2521 108 4 v 2543 2568 a(n)p 1972 2607 656 4 v 2258 2654 a FC(3)p 2258 2668 34 4 v 2258 2716 a(4)2301 2687 y Fv(n)1874 2905 y FD(=)1972 2849 y(4)p 1972 2886 42 4 v 1972 2962 a(3)2023 2756 y Fu(s)p 2106 2756 624 4 v 2116 2849 a FD(1)p 2116 2886 42 4 v 2116 2962 a(2)2182 2905 y(ln)2349 2849 y(1)p 2275 2886 191 4 v 2275 2962 a(1)18 b Fs(\000)g Fv(\015)2494 2905 y FD(+)2587 2849 y(ln)13 b Fv(n)p 2587 2886 133 4 v 2628 2962 a(n)2729 2905 y(:)p Black 505 w FD(\(84\))p Black 648 3253 a(By)30 b(the)h(assumption)d(that)i Fv(\015)j(<)27 b FD(0)p Fv(:)p FD(032704)g(and)k Fv(n)c(>)h FD(2)2440 3182 y Fu(p)p 2522 3182 522 4 v 2522 3253 a Fv(e)14 b FD(ln)f(\(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)i FD(1)p Fv(=\015)34 b FD(it)c(follo)n(ws)523 3353 y(that)22 b(Expression)d(\()1143 3353 y SDict begin H.S end 1143 3353 a Black FD(84)p Black 1226 3299 a SDict begin H.R end 1226 3299 a 1226 3353 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.84) cvn H.B /ANN pdfmark end 1226 3353 a FD(\))j(is)f(strictly)e(less)h(than)i (one.)f(Th)n(us,)h(w)n(e)f(can)g(substitute)h(Expression)523 3452 y(\()555 3452 y SDict begin H.S end 555 3452 a Black FD(84)p Black 638 3399 a SDict begin H.R end 638 3399 a 638 3452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.84) cvn H.B /ANN pdfmark end 638 3452 a FD(\))28 b(in)n(to)f(Equation)e (\()1257 3452 y SDict begin H.S end 1257 3452 a Black FD(83)p Black 1341 3399 a SDict begin H.R end 1341 3399 a 1341 3452 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.83) cvn H.B /ANN pdfmark end 1341 3452 a FD(\))j(and)f(deriv)n(e)1187 3738 y SDict begin H.S end 1187 3738 a 1187 3738 a SDict begin 12 H.A end 1187 3738 a 1187 3738 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.85) cvn /DEST pdfmark end 1187 3738 a FD(1)18 b(+)1340 3682 y Fv(\015)5 b FD(\()p Fv(n)19 b Fs(\000)f Fv(i)p FD(\))p 1340 3719 293 4 v 1379 3795 a(1)g Fs(\000)g Fv(a)1566 3807 y Ft(i)1665 3738 y Fs(\024)23 b FD(1)18 b(+)2258 3682 y Fv(\015)5 b(n)p 1906 3719 802 4 v 1906 3827 a FD(1)18 b Fs(\000)2059 3794 y FC(4)p 2059 3808 34 4 v 2059 3856 a(3)2102 3735 y Fu(q)p 2185 3735 523 4 v 2195 3794 a FC(1)p 2195 3808 34 4 v 2195 3856 a(2)2252 3827 y FD(ln)2390 3794 y FC(1)p 2345 3808 124 4 v 2345 3856 a(1)p Fq(\000)p Ft(\015)2497 3827 y FD(+)2590 3794 y FC(ln)11 b Ft(n)p 2590 3808 108 4 v 2623 3856 a(n)2717 3738 y Fv(:)517 b FD(\(85\))648 4118 y(By)33 b(substituting)f(the)i(upp)r(er)g(b)r(ound)g(in)f (Equation)f(\()2407 4118 y SDict begin H.S end 2407 4118 a Black FD(85)p Black 2490 4064 a SDict begin H.R end 2490 4064 a 2490 4118 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.85) cvn H.B /ANN pdfmark end 2490 4118 a FD(\))i(in)n(to)f(Inequalit)n(y)e(\()3161 4118 y SDict begin H.S end 3161 4118 a Black FD(82)p Black 3244 4064 a SDict begin H.R end 3244 4064 a 3244 4118 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.82) cvn H.B /ANN pdfmark end 3244 4118 a FD(\))j(w)n(e)523 4217 y(deriv)n(e)26 b(the)i(statemen)n(t)e(of)i(the)g(Lemma.)1522 b Fs(u)-55 b(t)523 4375 y SDict begin H.S end 523 4375 a 523 4375 a SDict begin 12 H.A end 523 4375 a 523 4375 a SDict begin [/View [/XYZ H.V]/Dest (lemma.19) cvn /DEST pdfmark end 523 4375 a Black 127 x Fr(Lemma)30 b(19.)p Black 40 w Fp(F)-6 b(or)30 b(any)g FD(1)23 b Fs(\024)f Fv(i)h Fs(\024)g Fv(n)p Fp(,)977 4821 y Fv(P)12 b(r)r(ob)1157 4679 y Fu(")1205 4821 y Fv(X)1274 4833 y Ft(i)1325 4821 y Fs(\025)1423 4764 y Fv(i)18 b Fs(\000)g FD(1)p 1423 4801 172 4 v 1483 4878 a Fv(n)1623 4821 y FD(+)1706 4671 y Fu(s)p 1789 4671 624 4 v 1799 4764 a FD(1)p 1799 4801 42 4 v 1799 4878 a(2)1864 4821 y(ln)2031 4764 y(1)p 1957 4801 191 4 v 1957 4878 a(1)g Fs(\000)g Fv(\015)2176 4821 y FD(+)2269 4764 y(ln)13 b Fv(n)p 2269 4801 133 4 v 2310 4878 a(n)2441 4675 y Fu(\014)2441 4725 y(\014)2441 4775 y(\014)2441 4825 y(\014)2441 4875 y(\014)2469 4821 y Fv(I)2505 4833 y Ft(i)2556 4821 y FD(=)23 b(1)2686 4679 y Fu(#)2757 4821 y Fs(\024)2877 4764 y FD(1)p 2854 4801 88 4 v 2854 4878 a Fv(n)2904 4854 y FC(2)p Black 1922 5173 a FD(43)p Black eop end end %%Page: 44 44 TeXDict begin HPSdict begin 44 43 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.44) cvn /DEST pdfmark end 515 125 a Black Black Black 523 464 a Fp(Pr)l(o)l(of.)p Black 43 w FD(De\014ne)33 b Fv(a)1089 476 y Ft(i)1149 464 y FD(=)1255 431 y Ft(i)p Fq(\000)p FC(1)p 1255 445 109 4 v 1289 492 a Ft(n)1395 464 y FD(+)1482 372 y Fu(q)p 1565 372 523 4 v 1575 431 a FC(1)p 1575 445 34 4 v 1575 492 a(2)1632 464 y FD(ln)1770 431 y FC(1)p 1725 445 124 4 v 1725 492 a(1)p Fq(\000)p Ft(\015)1876 464 y FD(+)1969 431 y FC(ln)11 b Ft(n)p 1969 445 108 4 v 2003 492 a(n)2087 464 y FD(.)33 b(F)-7 b(rom)31 b(Ba)n(y)n(es)h(rule)f(and)i(Lemmata)3322 464 y SDict begin H.S end 3322 464 a Black FD(14)p Black 3405 410 a SDict begin H.R end 3405 410 a 3405 464 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.14) cvn H.B /ANN pdfmark end 3405 464 a 523 584 a FD(and)684 584 y SDict begin H.S end 684 584 a Black FD(20)p Black 767 531 a SDict begin H.R end 767 531 a 767 584 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.20) cvn H.B /ANN pdfmark end 767 584 a 28 w FD(w)n(e)27 b(get)h(that)1095 795 y SDict begin H.S end 1095 795 a 1095 795 a SDict begin 12 H.A end 1095 795 a 1095 795 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.86) cvn /DEST pdfmark end 1095 795 a FD(Prob)1272 727 y Fu(\002)1320 795 y Fv(X)1389 807 y Ft(i)1440 795 y Fs(\025)23 b Fv(a)1572 724 y Fu(\014)1572 774 y(\014)1599 795 y Fv(I)1635 807 y Ft(i)1686 795 y FD(=)g(1)1830 727 y Fu(\003)1887 795 y FD(=)1985 739 y(Prob)o([)13 b Fv(X)2267 751 y Ft(i)2318 739 y Fs(\025)23 b Fv(a)2450 751 y Ft(i)2496 739 y Fs(^)18 b Fv(I)2605 751 y Ft(i)2657 739 y FD(=)k(1)14 b(])p 1985 776 838 4 v 2170 852 a(Prob)o([)g Fv(I)2420 864 y Ft(i)2471 852 y FD(=)23 b(1)14 b(])1887 1024 y Fs(\024)1985 968 y FD(Prob)o([)f Fv(X)2267 980 y Ft(i)2318 968 y Fs(\025)23 b Fv(a)2450 980 y Ft(i)2491 968 y FD(])p 1985 1005 530 4 v 2016 1081 a(Prob)o([)14 b Fv(I)2266 1093 y Ft(i)2317 1081 y FD(=)22 b(1)14 b(])1887 1266 y Fs(\024)2015 1172 y FC(1)p 1995 1186 74 4 v 1995 1234 a Ft(n)2036 1217 y Fb(2)2078 1205 y FD(\(1)19 b Fs(\000)f Fv(\015)5 b FD(\))2334 1175 y Ft(n)p 1985 1247 394 4 v 1989 1323 a FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2244 1299 y Ft(n)p Fq(\000)p FC(1)p Black 3257 1266 a FD(\(86\))p Black 1887 1487 a Fs(\024)2008 1431 y FD(1)p 1985 1468 88 4 v 1985 1544 a Fv(n)2035 1520 y FC(2)2110 1487 y Fv(:)523 1682 y FD(W)-7 b(e)26 b(use)g(Lemma)1101 1682 y SDict begin H.S end 1101 1682 a Black FD(20)p Black 1184 1629 a SDict begin H.R end 1184 1629 a 1184 1682 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.20) cvn H.B /ANN pdfmark end 1184 1682 a 26 w FD(to)g(b)r(ound)g(the)h(n)n(umerator)c(in)i(Inequalit)n(y)f(\()2626 1682 y SDict begin H.S end 2626 1682 a Black FD(86)p Black 2709 1629 a SDict begin H.R end 2709 1629 a 2709 1682 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.86) cvn H.B /ANN pdfmark end 2709 1682 a FD(\))i(and)g(Lemma)3222 1682 y SDict begin H.S end 3222 1682 a Black FD(14)p Black 3305 1629 a SDict begin H.R end 3305 1629 a 3305 1682 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.14) cvn H.B /ANN pdfmark end 3305 1682 a 26 w FD(to)523 1782 y(b)r(ound)i(the)g (denominator)d(in)h(Inequalit)n(y)f(\()1927 1782 y SDict begin H.S end 1927 1782 a Black FD(86)p Black 2011 1728 a SDict begin H.R end 2011 1728 a 2011 1782 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.86) cvn H.B /ANN pdfmark end 2011 1782 a FD(\).)1284 b Fs(u)-55 b(t)523 1862 y SDict begin H.S end 523 1862 a 523 1862 a SDict begin 12 H.A end 523 1862 a 523 1862 a SDict begin [/View [/XYZ H.V]/Dest (lemma.20) cvn /DEST pdfmark end 523 1862 a Black 79 x Fr(Lemma)30 b(20.)p Black 40 w Fp(F)-6 b(or)30 b FD(1)23 b Fs(\024)f Fv(i)h Fs(\024)g Fv(n)p Fp(,)936 2189 y FD(Prob)1113 2047 y Fu(")1175 2189 y Fv(X)1244 2201 y Ft(i)1295 2189 y Fs(\025)1392 2133 y Fv(i)18 b Fs(\000)h FD(1)p 1392 2170 172 4 v 1453 2246 a Fv(n)1592 2189 y FD(+)1675 2040 y Fu(s)p 1758 2040 624 4 v 1768 2133 a FD(1)p 1768 2170 42 4 v 1768 2246 a(2)1834 2189 y(ln)2001 2133 y(1)p 1927 2170 191 4 v 1927 2246 a(1)f Fs(\000)g Fv(\015)2146 2189 y FD(+)2239 2133 y(ln)13 b Fv(n)p 2239 2170 133 4 v 2280 2246 a(n)2425 2047 y Fu(#)2497 2189 y Fs(\024)2617 2133 y FD(1)p 2594 2170 88 4 v 2594 2246 a Fv(n)2644 2222 y FC(2)2691 2189 y FD(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2946 2155 y Ft(n)p Black 523 2431 a Fp(Pr)l(o)l(of.)p Black 43 w FD(The)26 b(probabilit)n(y)21 b(that)26 b Fv(X)1621 2443 y Ft(i)1672 2431 y Fs(\025)c Fv(p)k FD(for)f(some)f Fv(p)e Fs(2)i FD([0)p Fv(;)14 b FD(1\))25 b(is)f(exactly)g(the)i(probabilit)n(y)523 2531 y(that)i(for)f(some)f Fv(k)g Fs(\024)d Fv(i)18 b Fs(\000)g FD(1)27 b(exactly)f Fv(k)k FD(of)e(the)g(random)d(p)r(oin)n (ts)i(lie)f(in)h([0)p Fv(;)14 b(p)p FD(].)27 b(That)g(is)1279 2778 y SDict begin H.S end 1279 2778 a 1279 2778 a SDict begin 12 H.A end 1279 2778 a 1279 2778 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.87) cvn /DEST pdfmark end 1279 2778 a FD(Prob)o([)14 b Fv(X)1562 2790 y Ft(i)1612 2778 y Fs(\025)23 b Fv(p)14 b FD(])23 b(=)1896 2675 y Ft(i)p Fq(\000)p FC(1)1890 2700 y Fu(X)1889 2878 y Ft(k)q FC(=0)2024 2661 y Fu(\022)2085 2722 y Fv(n)2087 2835 y(k)2135 2661 y Fu(\023)2196 2778 y Fv(p)2238 2744 y Ft(i)2265 2778 y FD(\(1)c Fs(\000)f Fv(p)p FD(\))2515 2744 y Ft(n)p Fq(\000)p Ft(k)3257 2778 y FD(\(87\))523 3025 y(Notice)27 b(that)h(the)h(righ)n(t)d(hand)i(side)f(of)h(Equation) e(\()2170 3025 y SDict begin H.S end 2170 3025 a Black FD(87)p Black 2254 2972 a SDict begin H.R end 2254 2972 a 2254 3025 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.87) cvn H.B /ANN pdfmark end 2254 3025 a FD(\))i(is)f(exactly)f(the)j (probabilit)n(y)24 b(that)523 3125 y(a)29 b(Binomial)c(random)j(v)-5 b(ariable)26 b(with)j Fv(n)h FD(trials)d(and)i(success)g(probabilit)n (y)c Fv(p)30 b FD(\()p Fv(B)t(in)p FD(\()p Fv(n;)14 b(p)p FD(\)\))523 3224 y(is)27 b(at)g(most)f Fv(i)18 b Fs(\000)g FD(1.)28 b(So)f(w)n(e)g(can)g(apply)g(Ho)r(e\013ding's)f(inequalit)n(y) 2569 3224 y SDict begin H.S end 2569 3224 a -30 x FC(5)2606 3224 y SDict begin 12 H.L end 2606 3224 a 2606 3224 a SDict begin [/Subtype /Link/Dest (Hfootnote.5) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 2606 3224 a 28 w FD(to)h(get)902 3392 y(Prob)o([)14 b Fv(X)1185 3404 y Ft(i)1235 3392 y Fs(\025)23 b Fv(p)14 b FD(])23 b(=)f(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))23 b Fs(\024)f Fv(i)c Fs(\000)g FD(1)c(])1425 3566 y(=)22 b(Prob)1689 3449 y Fu(\024)1747 3566 y Fv(B)t(in)p FD(\()p Fv(n;)14 b(p)p FD(\))22 b Fs(\024)2196 3449 y Fu(\022)2257 3566 y Fv(p)c Fs(\000)2400 3449 y Fu(\022)2462 3566 y Fv(p)g Fs(\000)2615 3510 y Fv(i)g Fs(\000)g FD(1)p 2615 3547 172 4 v 2676 3623 a Fv(n)2796 3449 y Fu(\023\023)2918 3566 y Fv(n)2982 3449 y Fu(\025)1425 3776 y Fs(\024)k Fv(e)1551 3740 y Fq(\000)p FC(2)1636 3746 y FD(\()1668 3740 y Ft(p)p Fq(\000)1764 3715 y Fo(i)p Fd(\000)p Fb(1)p 1764 3728 96 3 v 1794 3761 a Fo(n)1870 3746 y FD(\))1902 3705 y Fb(2)1934 3740 y Ft(n)523 3962 y FD(By)27 b(c)n(ho)r(osing)e Fv(p)e FD(=)1153 3930 y Ft(i)p Fq(\000)p FC(1)p 1153 3944 109 4 v 1187 3991 a Ft(n)1290 3962 y FD(+)1373 3871 y Fu(q)p 1456 3871 523 4 v 1466 3930 a FC(1)p 1466 3944 34 4 v 1466 3991 a(2)1523 3962 y FD(ln)1661 3930 y FC(1)p 1616 3944 124 4 v 1616 3991 a(1)p Fq(\000)p Ft(\015)1768 3962 y FD(+)1861 3930 y FC(ln)10 b Ft(n)p 1861 3944 108 4 v 1894 3991 a(n)2006 3962 y FD(w)n(e)27 b(get)h(that)1386 4191 y(Prob)o([)14 b Fv(X)1669 4203 y Ft(i)1719 4191 y Fs(\025)23 b Fv(p)14 b FD(])23 b Fs(\024)g Fv(e)2036 4155 y Fq(\000)p FC(2)2121 4161 y FD(\()2153 4155 y Ft(p)p Fq(\000)2249 4129 y Fo(i)p Fd(\000)p Fb(1)p 2249 4142 96 3 v 2278 4175 a Fo(n)2354 4161 y FD(\))2386 4120 y Fb(2)2419 4155 y Ft(n)1909 4331 y FD(=)g Fv(e)2036 4294 y Fq(\000)p Ft(n)11 b FC(ln)2256 4272 y Fb(1)p 2216 4281 108 3 v 2216 4314 a(1)p Fd(\000)p Fo(\015)2333 4294 y Fq(\000)p FC(2)h(ln)f Ft(n)1909 4483 y FD(=)2007 4426 y(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))2262 4396 y Ft(n)p 2007 4464 301 4 v 2113 4540 a Fv(n)2163 4516 y FC(2)523 4673 y FD(\014nishing)26 b(the)i(pro)r(of)f(of)g(the)h(lemma.)1631 b Fs(u)-55 b(t)p Black 523 4737 473 4 v 546 4790 a Fn(5)606 4822 y SDict begin H.S end 606 4822 a 606 4822 a SDict begin H.R end 606 4822 a 606 4822 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.5) cvn /DEST pdfmark end 606 4822 a FB(Recall)84 b(that)e(Ho)r(e\013ding's)i(inequalit)n(y)e(for)i (a)f(Binomial)h(random)e(v)l(ariable)h(is)606 4924 y(Prob[)14 b Fy(B)t(in)p FB(\()p Fy(n;)f(p)p FB(\))21 b Fk(\024)g FB(\()p Fy(p)c Fk(\000)f Fy(\017)p FB(\))p Fy(n)d FB(])22 b Fk(\024)f Fy(e)1661 4893 y Fi(\000)p Fn(2)p Fj(\017)1765 4872 y Fb(2)1798 4893 y Fj(n)1840 4924 y FB(.)26 b(W)-6 b(e)25 b(use)h(this)g(with)g Fy(\017)21 b FB(=)g Fy(p)c Fk(\000)2753 4894 y Fj(i)p Fi(\000)p Fn(1)p 2753 4908 101 4 v 2784 4950 a Fj(n)2863 4924 y FB(.)p Black Black 1922 5173 a FD(44)p Black eop end end %%Page: 45 45 TeXDict begin HPSdict begin 45 44 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.45) cvn /DEST pdfmark end 515 125 a Black Black 523 365 a SDict begin H.S end 523 365 a 523 365 a SDict begin 12 H.A end 523 365 a 523 365 a SDict begin [/View [/XYZ H.V]/Dest (lemma.21) cvn /DEST pdfmark end 523 365 a Black 83 x Fr(Lemma)30 b(21.)p Black 40 w Fp(F)-6 b(or)32 b(any)g FD(0)25 b Fs(\024)h Fv(i)g Fs(\024)g Fv(n)p Fp(,)31 b(c)l(onditione)l(d)i(on)f(the)g(event)f Fv(X)2744 460 y Ft(i)2798 448 y FD(=)25 b Fv(x)p Fp(,)33 b(the)e(r)l(andom)523 548 y(variables)h Fv(C)927 560 y Ft(i)984 548 y Fp(and)f Fv(I)1182 560 y Ft(i)1240 548 y Fp(ar)l(e)f(indep)l(endent.)h(F)-6 b(ormal)t(ly,)937 733 y FD(Prob)n([)14 b Fv(C)1209 745 y Ft(i)1260 733 y Fs(\024)23 b Fv(k)j Fs(j)d FD(\()p Fv(X)1564 745 y Ft(i)1615 733 y FD(=)f Fv(x)p FD(\))d Fs(^)g Fv(I)1910 745 y Ft(i)1952 733 y FD(])k(=)g(Prob)o([)14 b Fv(C)2359 745 y Ft(i)2410 733 y Fs(\024)22 b Fv(k)k Fs(j)d FD(\()p Fv(X)2713 745 y Ft(i)2764 733 y FD(=)g Fv(x)p FD(\))14 b(])p Fv(:)p Black 523 918 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Giv)n(en)21 b(that)i Fv(X)1262 930 y Ft(i)1312 918 y FD(=)g Fv(x)p FD(,)g(the)f(set)h Fv(A)f FD(of)h(the)f Fv(i)g FD(items)f(arriving)d(at)k(times)f Fs(\024)i Fv(x)p FD(,)g(and)f(the)523 1018 y(arriv)-5 b(al)27 b(times)h(of)i(the)g (items)f(in)g Fv(A)i FD(\(that)f(is)f Fv(X)2039 1030 y FC(1)2076 1018 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)2330 1030 y Ft(i)p Fq(\000)p FC(1)2442 1018 y FD(\),)31 b(the)f(action)f(of)h (the)g(greedy)523 1117 y(algorithm)22 b(on)k(the)h Fv(i)p FD(th)g(item)e(arriving)d(at)k Fv(x)h FD(is)f(determined)e(\(that)j(is) f Fv(I)2807 1129 y Ft(i)2861 1117 y FD(is)f(determined\).)523 1217 y(On)i(the)h(other)f(hand,)h Fv(C)1311 1229 y Ft(i)1367 1217 y FD(dep)r(ends)g(only)e(on)h(the)h(arriv)-5 b(al)24 b(times)i(of)i(the)g(p)r(oin)n(ts)e(not)i(in)f Fv(A)523 1317 y FD(\(that)g(is)e Fv(Z)c Fs(n)15 b Fv(A)p FD(\).)26 b(Since)g(this)f(holds)g(for)g(an)n(y)h(set)g Fv(A)p FD(,)g(and)g(for)g(an)n(y)f(arriv)-5 b(al)23 b(times)h(of)i(these)523 1416 y(p)r(oin)n(ts,)h(it)g(also)e(holds)h(without)h(conditioning)d(on) k Fv(A)p FD(.)1103 b Fs(u)-55 b(t)523 1510 y SDict begin H.S end 523 1510 a 523 1510 a SDict begin 12 H.A end 523 1510 a 523 1510 a SDict begin [/View [/XYZ H.V]/Dest (lemma.22) cvn /DEST pdfmark end 523 1510 a Black 85 x Fr(Lemma)30 b(22.)p Black 40 w Fp(F)-6 b(or)30 b(any)g FD(0)23 b Fs(\024)f Fv(i)h Fs(\024)g Fv(n)p Fp(,)1348 1796 y Fv(E)5 b FD([)p Fv(C)1496 1808 y Ft(i)1547 1796 y Fs(j)24 b Fv(X)1663 1808 y Ft(i)1713 1796 y FD(=)f Fv(x)p FD(])g Fs(\024)g FD(1)18 b(+)2206 1740 y Fv(\015)p 2135 1777 191 4 v 2135 1853 a FD(1)g Fs(\000)g Fv(x)2335 1796 y FD(\()p Fv(n)h Fs(\000)f Fv(i)p FD(\))p Black 523 2012 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Conditioning)26 b(on)j Fv(X)1469 2024 y Ft(i)1523 2012 y FD(=)c Fv(x)p FD(,)30 b(there)g(are)e(exactly)g Fv(n)19 b Fs(\000)h Fv(i)29 b FD(p)r(oin)n(ts)f(eac)n(h)h(distributed)523 2111 y(uniformly)k(in)i([)p Fv(x;)14 b FD(1\).)36 b(Let)g Fs(f)p Fv(Z)1511 2123 y FC(1)1547 2111 y Fv(;)14 b(:)g(:)g(:)g(Z)1752 2123 y Ft(n)p Fq(\000)p Ft(i)1872 2111 y Fs(g)35 b FD(b)r(e)h Fv(n)24 b Fs(\000)f Fv(i)36 b FD(indep)r(enden)n(t)g(random)d(v)-5 b(ariables)523 2211 y(eac)n(h)29 b(distributed)f(uniformly)e(in)j([)p Fv(x;)14 b FD(1\).)30 b(Since)f Fv(C)2130 2223 y Ft(i)2188 2211 y FD(con)n(tains)e(the)j Fv(i)p FD(th)g(p)r(oin)n(t)f(and)g(an)n (y)g(of)523 2311 y(the)f(follo)n(wing)c(p)r(oin)n(ts)i(the)i(falls)d (in)i([)p Fv(x;)14 b(x)20 b FD(+)e Fv(\015)5 b FD(\))27 b(w)n(e)g(ha)n(v)n(e)g(that)1242 2495 y(E[)o Fv(C)1380 2507 y Ft(i)1431 2495 y Fs(j)c Fv(X)1546 2507 y Ft(i)1597 2495 y FD(=)f Fv(x)p FD(])i(=)f(1)18 b(+)g(E[)o Fs(j)q(f)p Fv(Z)2211 2507 y Ft(i)2260 2495 y Fv(<)23 b(\015)5 b Fs(g)o(j)p FD(])1778 2700 y(=)23 b(1)18 b(+)2011 2596 y Ft(n)p Fq(\000)p Ft(i)2009 2621 y Fu(X)2015 2798 y Ft(i)p FC(=1)2143 2700 y FD(Prob)n([)c Fv(Z)2413 2712 y Ft(i)2464 2700 y Fv(<)22 b(\015)d FD(])27 b Fv(:)523 2961 y FD(If)h Fv(x)c Fs(\024)e FD(1)c Fs(\000)g Fv(\015)32 b FD(then)d(Prob)n([)14 b Fv(Z)1442 2973 y Ft(i)1492 2961 y Fv(<)23 b(\015)c FD(])k(=)1827 2924 y Ft(\015)p 1785 2942 123 4 v 1785 2990 a FC(1)p Fq(\000)p Ft(x)1918 2961 y FD(,)28 b(otherwise)d(Prob)o([)14 b Fv(Z)2609 2973 y Ft(i)2659 2961 y Fv(<)23 b(\015)18 b FD(])23 b(=)g(1,)k (therefore)1239 3241 y(E[)o Fv(C)1377 3253 y Ft(i)1428 3241 y Fs(j)c Fv(X)1543 3253 y Ft(i)1594 3241 y FD(=)g Fv(x)p FD(])g(=)g(1)18 b(+)2008 3137 y Ft(n)p Fq(\000)p Ft(i)2006 3162 y Fu(X)2012 3339 y Ft(i)p FC(=1)2140 3241 y FD(min)2278 3124 y Fu(\022)2339 3241 y FD(1)p Fv(;)2499 3185 y(\015)p 2428 3222 191 4 v 2428 3298 a FD(1)g Fs(\000)g Fv(x)2628 3124 y Fu(\023)1775 3531 y Fs(\024)23 b FD(1)18 b(+)2008 3427 y Ft(n)p Fq(\000)p Ft(i)2006 3452 y Fu(X)2012 3629 y Ft(i)p FC(=1)2221 3475 y Fv(\015)p 2150 3512 V 2150 3588 a FD(1)g Fs(\000)g Fv(x)1775 3789 y FD(=)23 b(1)18 b(+)2016 3733 y Fv(\015)5 b FD(\()p Fv(n)18 b Fs(\000)g Fv(i)p FD(\))p 2016 3770 293 4 v 2067 3846 a(1)g Fs(\000)g Fv(x)3350 3992 y Fs(u)-55 b(t)523 4070 y SDict begin H.S end 523 4070 a 523 4070 a SDict begin 12 H.A end 523 4070 a 523 4070 a SDict begin [/View [/XYZ H.V]/Dest (lemma.23) cvn /DEST pdfmark end 523 4070 a Black 100 x Fr(Lemma)30 b(23.)p Black 40 w Fp(F)-6 b(or)30 b(al)t(l)h Fv(i;)14 b(k)25 b Fs(\025)e FD(1)29 b Fp(such)h(that)g Fv(i)18 b FD(+)g Fv(k)26 b Fs(\024)c Fv(n)p Fp(,)1015 4355 y FD(E[)o Fv(m)p FD(\()p Fs(f)p Fv(X)1310 4367 y FC(1)1347 4355 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)1601 4367 y Ft(k)1641 4355 y Fs(g)p Fv(;)g(\015)5 b FD(\)])23 b(=)f(E[)p Fv(m)p FD(\()p Fs(f)p Fv(X)2229 4367 y Ft(i)p FC(+1)2340 4355 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)2594 4367 y Ft(i)p FC(+)p Ft(k)2709 4355 y Fs(g)p Fv(;)g(\015)5 b FD(\)])o Fv(:)p Black 523 4540 a Fp(Pr)l(o)l(of.)p Black 43 w FD(The)26 b(n)n(um)n(b)r(er)e(of)i Fv(\015)5 b FD(-indep)r(enden)n(t)25 b(p)r(oin)n(ts)f(are)h(in)n(v)-5 b(arian)n(t)22 b(under)k(translation)c(and)523 4640 y(rotation,)i Fw(i.e.)p FD(,)i(for)g(an)n(y)f(real)g Fv(x)i FD(and)f(for)g(ev)n(ery)f Fv(`)p FD(,)h(1)c Fs(\024)h Fv(`)g Fs(\024)f Fv(n)p FD(,)27 b(and)f(for)g(an)n(y)f(set)i(of)f(p)r(oin)n(ts)523 4740 y Fs(f)p Fv(x)612 4752 y FC(1)649 4740 y Fv(;)14 b(:)g(:)g(:)g(;)g(x) 881 4752 y Ft(`)913 4740 y Fs(g)p FD(,)1095 4924 y SDict begin H.S end 1095 4924 a 1095 4924 a SDict begin 12 H.A end 1095 4924 a 1095 4924 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.88) cvn /DEST pdfmark end 1095 4924 a Fv(m)p FD(\()p Fs(f)p Fv(x)1289 4936 y FC(1)1326 4924 y Fv(;)g(:)g(:)g(:)g(;)g(x)1558 4936 y Ft(`)1590 4924 y Fs(g)p Fv(;)g(\015)5 b FD(\))22 b(=)h Fv(m)p FD(\()p Fs(f)p Fv(x)18 b Fs(\000)g Fv(x)2201 4936 y FC(1)2239 4924 y Fv(;)c(:)g(:)g(:)g(;)g(x)19 b Fs(\000)f Fv(x)2620 4936 y Ft(`)2652 4924 y Fs(g)p Fv(;)c(\015)5 b FD(\))p Fv(:)423 b FD(\(88\))p Black 1922 5173 a(45)p Black eop end end %%Page: 46 46 TeXDict begin HPSdict begin 46 45 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.46) cvn /DEST pdfmark end 515 125 a Black Black 648 448 a FD(Assume)18 b(that)h Fv(i)r FD(+)r Fv(k)25 b FD(=)d Fv(n)p FD(.)e(The)f(v)n(ectors)e(\()p Fv(X)2000 460 y FC(1)2038 448 y Fv(;)d(:)g(:)g(:)f(;)h(X)2291 460 y Ft(k)2332 448 y FD(\))19 b(and)g(\(1)r Fs(\000)r Fv(X)2748 460 y Ft(n)2792 448 y Fv(;)14 b(:)g(:)g(:)g(;)g FD(1)r Fs(\000)r Fv(X)3157 460 y Ft(n)p Fq(\000)p Ft(k)q FC(+1)3373 448 y FD(\))523 548 y(ha)n(v)n(e)26 b(the)i(same)e (distribution,)f(and)j(the)g(lemma)c(follo)n(ws)h(from)h(\()2614 548 y SDict begin H.S end 2614 548 a Black FD(88)p Black 2697 494 a SDict begin H.R end 2697 494 a 2697 548 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.88) cvn H.B /ANN pdfmark end 2697 548 a FD(\).)648 648 y(Otherwise,)e(\()p Fv(i)15 b FD(+)h Fv(k)26 b(<)c(n)p FD(\),)27 b(w)n(e)e(condition)f(on)i Fv(X)2173 660 y Ft(i)p FC(+)p Ft(k)q FC(+1)2396 648 y FD(=)c Fv(x)p FD(.)27 b(It)f(no)n(w)g(follo)n(ws)d(that)k(the)523 747 y(v)n(ectors)h(\()p Fv(X)908 759 y FC(1)945 747 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)1199 759 y Ft(k)1239 747 y FD(\))30 b(and)f(\()p Fv(x)20 b Fs(\000)f Fv(X)1716 759 y Ft(i)p FC(+)p Ft(k)1832 747 y Fv(;)14 b(:)g(:)g(:)f(;)h(x)20 b Fs(\000)f Fv(X)2236 759 y Ft(i)p FC(+1)2348 747 y FD(\))29 b(ha)n(v)n(e)f(the)i(same)e(distribution.)523 847 y(Therefore,)e(it)h (follo)n(ws)e(from)h(\()1506 847 y SDict begin H.S end 1506 847 a Black FD(88)p Black 1590 793 a SDict begin H.R end 1590 793 a 1590 847 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.88) cvn H.B /ANN pdfmark end 1590 847 a FD(\))i(that)523 999 y(E[)p Fv(m)p FD(\()p Fs(f)p Fv(X)819 1011 y FC(1)855 999 y Fv(;)14 b(:)g(:)g(:)g(;)g(X)1109 1011 y Ft(k)1149 999 y Fs(g)p Fv(;)g(\015)5 b FD(\))23 b Fs(j)g Fv(X)1446 1011 y Ft(i)p FC(+)p Ft(k)q FC(+1)1668 999 y FD(=)g Fv(x)p FD(])g(=)g(E[)p Fv(m)p FD(\()p Fs(f)p Fv(X)2233 1011 y Ft(i)p FC(+1)2344 999 y Fv(;)14 b(:)g(:)g(:)f(;)h(X)2597 1011 y Ft(i)p FC(+)p Ft(k)2712 999 y Fs(g)p Fv(;)g(\015)5 b FD(\))23 b Fs(j)g Fv(X)3009 1011 y Ft(i)p FC(+)p Ft(k)q FC(+1)3231 999 y FD(=)g Fv(x)p FD(])p Fv(:)523 1151 y FD(Since)k(the)h(ab)r(o)n(v)n(e)e(holds)g(for)h(ev)n(ery)g Fv(x)p FD(,)h(the)g(lemma)c(follo)n(ws.)889 b Fs(u)-55 b(t)648 1294 y FD(W)-7 b(e)20 b(no)n(w)f(giv)n(e)f(sev)n(eral)g(tec)n (hnical)f(lemmata)g(required)h(to)i(conclude)f(the)h(pro)r(ofs)f(ab)r (o)n(v)n(e.)523 1394 y SDict begin H.S end 523 1394 a 523 1394 a SDict begin 12 H.A end 523 1394 a 523 1394 a SDict begin [/View [/XYZ H.V]/Dest (lemma.24) cvn /DEST pdfmark end 523 1394 a Black 133 x Fr(Lemma)30 b(24.)p Black 40 w Fp(F)-6 b(or)30 b(any)g FD(0)23 b Fv(<)f(\013)i(<)f FD(1)1530 1713 y(\(1)18 b Fs(\000)g Fv(\013)p FD(\))1790 1679 y FC(1)p Ft(=\013)1928 1713 y Fs(\025)2026 1657 y FD(1)p 2026 1694 42 4 v 2027 1770 a Fv(e)2096 1713 y Fs(\001)g FD(\(1)g Fs(\000)h Fv(\013)p FD(\))p Black 523 1893 a Fp(Pr)l(o)l(of.)p Black 43 w FD(F)-7 b(or)25 b(an)n(y)h Fv(x)g FD(suc)n(h)g(that)g(1)15 b Fs(\000)f Fv(\013)24 b Fs(\024)e Fv(x)i Fs(\024)e FD(1)k(w)n(e)g(ha)n(v)n(e)e (that)j(1)22 b Fs(\024)h FD(1)p Fv(=x)f Fs(\024)h FD(1)p Fv(=)p FD(\(1)14 b Fs(\000)h Fv(\013)p FD(\).)26 b(So)523 1993 y(b)n(y)h(in)n(tegrating)e(from)h(1)18 b Fs(\000)g Fv(\013)28 b FD(to)f(1)h(w)n(e)f(get)g(that)1827 2170 y Fu(Z)1910 2190 y FC(1)1874 2358 y(1)p Fq(\000)p Ft(\013)2019 2227 y FD(1)p 2016 2264 48 4 v 2016 2340 a Fv(x)2087 2283 y FD(d)p Fv(x)d Fs(\024)2291 2170 y Fu(Z)2374 2190 y FC(1)2337 2358 y(1)p Fq(\000)p Ft(\013)2470 2283 y FD(1)p Fv(=)p FD(\(1)17 b Fs(\000)h Fv(\013)p FD(\))c(d)p Fv(x)1007 2468 y Fs(\))484 b FD(ln)o(\(1\))18 b Fs(\000)g FD(ln\(1)g Fs(\000)g Fv(\013)p FD(\))24 b Fs(\024)e Fv(\013=)p FD(\(1)c Fs(\000)g Fv(\013)p FD(\))1007 2641 y Fs(\))760 b FD(ln\(1)18 b Fs(\000)g Fv(\013)p FD(\))24 b Fs(\025)e(\000)2485 2585 y FD(1)p 2366 2622 280 4 v 2366 2698 a(1)p Fv(=\013)c Fs(\000)g FD(1)1007 2841 y Fs(\))812 b Fv(e)1941 2807 y FC(ln)o(\(1)p Fq(\000)p Ft(\013)p FC(\))2204 2841 y Fs(\025)22 b Fv(e)2330 2801 y Fq(\000)2463 2779 y Fb(1)p 2392 2788 170 3 v 2392 2822 a(1)p Fo(=\013)p Fd(\000)p Fb(1)1007 2993 y Fs(\))830 b FD(\(1)18 b Fs(\000)g Fv(\013)p FD(\))24 b Fs(\025)e Fv(e)2330 2953 y Fq(\000)2463 2931 y Fb(1)p 2392 2940 V 2392 2974 a(1)p Fo(=\013)p Fd(\000)p Fb(1)3257 2993 y SDict begin H.S end 3257 2993 a 3257 2993 a SDict begin 15 H.A end 3257 2993 a 3257 2993 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.89) cvn /DEST pdfmark end 3257 2993 a FD(\(89\))1007 3166 y Fs(\))630 b FD(\(1)18 b Fs(\000)g Fv(\013)p FD(\))1980 3132 y FC(1)p Ft(=\013)p Fq(\000)p FC(1)2204 3166 y Fs(\025)2301 3110 y FD(1)p 2301 3147 42 4 v 2302 3223 a Fv(e)3257 3166 y SDict begin H.S end 3257 3166 a 3257 3166 a SDict begin 15 H.A end 3257 3166 a 3257 3166 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.90) cvn /DEST pdfmark end 3257 3166 a FD(\(90\))1007 3366 y Fs(\))715 b FD(\(1)18 b Fs(\000)g Fv(\013)p FD(\))2065 3332 y FC(1)p Ft(=\013)2204 3366 y Fs(\025)2301 3310 y FD(1)p 2301 3347 42 4 v 2302 3423 a Fv(e)2371 3366 y Fs(\001)h FD(\(1)f Fs(\000)g Fv(\013)p FD(\))28 b Fv(:)3257 3366 y SDict begin H.S end 3257 3366 a 3257 3366 a SDict begin 15 H.A end 3257 3366 a 3257 3366 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.91) cvn /DEST pdfmark end 3257 3366 a FD(\(91\))523 3546 y(Inequalit)n(y)j(\()953 3546 y SDict begin H.S end 953 3546 a Black FD(90)p Black 1036 3492 a SDict begin H.R end 1036 3492 a 1036 3546 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.90) cvn H.B /ANN pdfmark end 1036 3546 a FD(\))j(follo)n(ws)d(from)h(\()1615 3546 y SDict begin H.S end 1615 3546 a Black FD(89)p Black 1699 3492 a SDict begin H.R end 1699 3492 a 1699 3546 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.89) cvn H.B /ANN pdfmark end 1699 3546 a FD(\))h(b)n(y)h(taking)e(the)i(\(1)p Fv(=\013)22 b Fs(\000)g FD(1\))33 b(p)r(o)n(w)n(er)g(of)g(b)r(oth)h (sides)523 3645 y(and)27 b(\()716 3645 y SDict begin H.S end 716 3645 a Black FD(91)p Black 800 3592 a SDict begin H.R end 800 3592 a 800 3645 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.91) cvn H.B /ANN pdfmark end 800 3645 a FD(\))h(follo)n(ws)c(from)j(\()1361 3645 y SDict begin H.S end 1361 3645 a Black FD(90)p Black 1444 3592 a SDict begin H.R end 1444 3592 a 1444 3645 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.90) cvn H.B /ANN pdfmark end 1444 3645 a FD(\))h(b)n(y)f(m)n(ultiplying)c(b)r(oth)28 b(sides)e(b)n(y)i(\(1)18 b Fs(\000)g Fv(\013)p FD(\).)494 b Fs(u)-55 b(t)523 3710 y SDict begin H.S end 523 3710 a 523 3710 a SDict begin 12 H.A end 523 3710 a 523 3710 a SDict begin [/View [/XYZ H.V]/Dest (lemma.25) cvn /DEST pdfmark end 523 3710 a Black 79 x Fr(Lemma)30 b(25.)p Black 40 w Fp(F)-6 b(or)30 b Fv(x)24 b Fs(\025)e FD(1)p Fv(=e)p Fp(,)1710 3825 y Fu(p)p 1793 3825 425 4 v 75 x Fv(e)d Fs(\001)f Fv(x)c FD(ln\(1)p Fv(=x)p FD(\))523 4031 y Fp(is)30 b(monotonic)l(al)t(ly)i(de)l(cr)l(e)l(asing,)f(and)f (for)h(any)f Fv(x)23 b(>)g FD(0)p Fp(,)1634 4108 y Fu(p)p 1717 4108 V 75 x Fv(e)18 b Fs(\001)h Fv(x)14 b FD(ln)o(\(1)p Fv(=x)p FD(\))24 b Fs(\024)e FD(1)p Black 523 4335 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Let)31 b Fv(f)9 b FD(\()p Fv(x)p FD(\))29 b(=)e Fv(x)14 b FD(ln\(1)p Fv(=x)p FD(\).)31 b(Then)f Fv(f)1866 4304 y Fq(0)1889 4335 y FD(\()p Fv(x)p FD(\))f(=)f Fs(\000)14 b FD(ln)e Fv(x)21 b Fs(\000)f FD(1)30 b(whic)n(h)g(is)g(p)r(ositiv)n(e)e(for)i(an)n(y)523 4434 y(0)37 b Fv(<)h(x)g(<)f FD(1)p Fv(=e)e FD(and)h(negativ)n(e)f(for) g(an)n(y)h Fv(x)i(>)f FD(1)p Fv(=e)p FD(.)f(Therefore)f(the)i(maxim)n (um)32 b(of)k Fv(f)45 b FD(is)523 4534 y(obtained)26 b(at)i Fv(x)23 b FD(=)g(1)p Fv(=e)p FD(.)j(It)i(follo)n(ws)d(that)j (for)f(all)e Fv(x)f(>)e FD(0)1369 4677 y Fu(p)p 1452 4677 V 75 x Fv(e)c Fs(\001)h Fv(x)14 b FD(ln)o(\(1)p Fv(=x)p FD(\))23 b Fs(\024)1987 4622 y Fu(r)p 2070 4622 287 4 v 130 x Fv(e)2119 4696 y FD(1)p 2119 4733 42 4 v 2120 4809 a Fv(e)2184 4752 y FD(ln)o(\()p Fv(e)p FD(\))g(=)g(1)k Fv(:)3350 4924 y Fs(u)-55 b(t)523 4968 y SDict begin H.S end 523 4968 a 523 4968 a SDict begin 12 H.A end 523 4968 a 523 4968 a SDict begin [/View [/XYZ H.V]/Dest (lemma.26) cvn /DEST pdfmark end 523 4968 a Black 1922 5173 a FD(46)p Black eop end end %%Page: 47 47 TeXDict begin HPSdict begin 47 46 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.47) cvn /DEST pdfmark end 515 125 a Black Black Black 523 448 a Fr(Lemma)30 b(26.)p Black 40 w Fp(F)-6 b(or)30 b(any)g Fv(n)23 b Fs(\025)g FD(2)29 b Fp(and)h(any)g FD(0)23 b Fv(<)g(\015)k Fs(\024)c FD(0)p Fv(:)p FD(5)1625 649 y(\(1)18 b Fs(\000)g Fv(\015)5 b FD(\))23 b Fs(\025)2100 593 y FD(1)p 2000 630 241 4 v 2000 706 a(1)18 b(+)g Fv(\015)5 b(n)2280 649 y(:)p Black 523 855 a Fp(Pr)l(o)l(of.)p Black 43 w FD(It)34 b(su\016ces)f(to)g(pro) n(v)n(e)f(the)i(ab)r(o)n(v)n(e)e(inequalit)n(y)e(for)i Fv(n)h FD(=)f(2,)h(for)g Fv(n)g(>)f FD(2)h(the)h(righ)n(t)523 954 y(hand)28 b(side)e(can)h(only)f(decrease)h(whereas)f(the)i(left)f (hand)h(side)e(do)r(es)i(not)f(dep)r(end)h(on)g Fv(n)p FD(.)648 1054 y(No)n(w,)g(\(1)19 b Fs(\000)g Fv(\015)5 b FD(\)\(1)19 b(+)g(2)p Fv(\015)5 b FD(\))25 b(=)g(1)18 b(+)h Fv(\015)24 b Fs(\000)19 b FD(2)p Fv(\015)1917 1024 y FC(2)1954 1054 y FD(,)29 b(and)f Fv(\015)c Fs(\000)19 b FD(2)p Fv(\015)2409 1024 y FC(2)2471 1054 y Fs(\025)25 b FD(0)j(for)h(all)d(0)f Fv(<)g(\015)30 b Fs(\024)25 b FD(0)p Fv(:)p FD(5)j(so)523 1153 y(the)g(lemma)d(holds.)2202 b Fs(u)-55 b(t)523 1212 y SDict begin H.S end 523 1212 a 523 1212 a SDict begin 12 H.A end 523 1212 a 523 1212 a SDict begin [/View [/XYZ H.V]/Dest (lemma.27) cvn /DEST pdfmark end 523 1212 a Black 99 x Fr(Lemma)30 b(27.)p Black 40 w Fp(F)-6 b(or)30 b(any)g FD(0)23 b Fv(<)f(\015)28 b(<)23 b FD(1)p Fv(=e)28 b Fp(we)i(have)h(that)1026 1471 y Fv(n)14 b FD(ln)f Fv(n)p 933 1508 384 4 v 933 1584 a FD(\()p Fv(n)18 b FD(+)g(1)p Fv(=\015)5 b FD(\))1280 1560 y FC(2)1350 1527 y Fs(\024)1437 1410 y Fu(\022)1508 1471 y Fv(\014)t FD(\()p Fv(\015)g FD(\))p 1508 1508 164 4 v 1569 1584 a(4)1682 1410 y Fu(\023)1761 1527 y Fs(\001)19 b Fv(\015)f FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b(+)2244 1410 y Fu(\022)2315 1471 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln)f(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2315 1508 489 4 v 2538 1584 a(4)2813 1410 y Fu(\023)2893 1527 y Fs(\001)18 b Fv(\015)5 b(;)523 1751 y Fp(wher)l(e)30 b Fv(\014)t FD(\()p Fv(\015)5 b FD(\))31 b Fp(is)f(de\014ne)l(d)g(to)g(b)l(e)f FD(1)18 b(+)1812 1718 y FC(2)p 1674 1732 310 4 v 1674 1780 a(ln)11 b(\(1)p Ft(=\015)t FC(\))p Fq(\000)p FC(1)1994 1751 y Fp(.)p Black 523 1934 a(Pr)l(o)l(of.)p Black 43 w FD(Fix)30 b(0)f Fv(<)f(\015)34 b(<)28 b FD(1)p Fv(=e)p FD(,)i(de\014ne)i Fv(h)p FD(\()p Fv(n)p FD(\))d(=)2044 1901 y Ft(n)12 b FC(ln)e Ft(n)p 1983 1915 282 4 v 1983 1963 a FC(\()p Ft(n)p FC(+1)p Ft(=\015)t FC(\))2233 1946 y Fb(2)2306 1934 y FD(and)31 b(let)g Fv(n)2645 1946 y FC(0)2711 1934 y FD(=)d(argmax)3076 1954 y Ft(n)p Fq(\025)p FC(1)3220 1934 y Fv(h)p FD(\()p Fv(n)p FD(\).)523 2048 y(Note)g(that)g Fv(\014)t FD(\()p Fv(\015)5 b FD(\))23 b Fv(>)g FD(1)k(for)g(0)c Fv(<)f(\015)28 b(<)23 b FD(1)p Fv(=e)p FD(,)j(w)n(e)h(sho)n(w)g(b)r (elo)n(w)f(that)1593 2214 y SDict begin H.S end 1593 2214 a 1593 2214 a SDict begin 12 H.A end 1593 2214 a 1593 2214 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.92) cvn /DEST pdfmark end 1593 2214 a FD(1)p Fv(=\015)g Fs(\024)d Fv(n)1884 2226 y FC(0)1944 2214 y Fs(\024)g Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p Fv(=\015)32 b(:)922 b FD(\(92\))523 2389 y(Assume)26 b(that)i(Equation)d(\()1400 2389 y SDict begin H.S end 1400 2389 a Black FD(92)p Black 1484 2335 a SDict begin H.R end 1484 2335 a 1484 2389 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.92) cvn H.B /ANN pdfmark end 1484 2389 a FD(\))i(holds.)f(As)i(b)r(oth)f Fv(n)14 b FD(ln)f Fv(n)27 b FD(and)h(\()p Fv(n)18 b FD(+)g(1)p Fv(=\015)5 b FD(\))2833 2347 y FC(2)2898 2389 y FD(are)26 b(monotoni-)523 2488 y(cally)f(increasing)f(in)j Fv(n)p FD(,)h(it)f(follo)n(ws)d(that)k(for)f(all)e Fv(n)e Fs(\025)g FD(1)1179 2654 y SDict begin H.S end 1179 2654 a 1179 2654 a SDict begin 12 H.A end 1179 2654 a 1179 2654 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.3.93) cvn /DEST pdfmark end 1179 2654 a Fv(h)p FD(\()p Fv(n)p FD(\))h Fs(\024)f Fv(h)p FD(\()p Fv(n)1583 2666 y FC(0)1620 2654 y FD(\))1365 2827 y(=)1512 2771 y Fv(n)1562 2783 y FC(0)1613 2771 y FD(ln)o(\()p Fv(n)1764 2783 y FC(0)1802 2771 y FD(\))p 1463 2808 421 4 v 1463 2884 a(\()p Fv(n)1545 2896 y FC(0)1600 2884 y FD(+)c(1)p Fv(=\015)5 b FD(\))1848 2860 y FC(2)1365 3056 y Fs(\024)1463 3000 y FD(\()p Fv(\014)t FD(\()p Fv(\015)g FD(\))p Fv(=\015)g FD(\))14 b(ln)o(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p Fv(=\015)g FD(\))p 1463 3037 718 4 v 1589 3113 a(\(1)p Fv(=\015)22 b FD(+)c(1)p Fv(=\015)5 b FD(\))2017 3089 y FC(2)p Black 3257 3056 a FD(\(93\))p Black 1365 3287 a(=)1453 3170 y Fu(\022)1524 3231 y Fv(\014)t FD(\()p Fv(\015)g FD(\))p Fv(\015)p 1524 3268 212 4 v 1609 3344 a FD(4)1745 3170 y Fu(\023)1820 3287 y FD(ln)o(\()p Fv(\014)t FD(\()p Fv(\015)g FD(\))p Fv(=\015)g FD(\))1365 3518 y(=)1463 3462 y Fv(\014)t FD(\()p Fv(\015)g FD(\))p 1463 3499 164 4 v 1524 3575 a(4)1655 3518 y Fs(\001)18 b Fv(\015)h FD(ln)o(\(1)p Fv(=\015)5 b FD(\))17 b(+)2133 3462 y Fv(\014)t FD(\()p Fv(\015)5 b FD(\))14 b(ln\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\)\))p 2133 3499 475 4 v 2350 3575 a(4)2636 3518 y Fs(\001)19 b Fv(\015)5 b(;)p Black 508 w FD(\(94\))p Black 648 3712 a(where)18 b(Inequalit)n(y)f(\()1295 3712 y SDict begin H.S end 1295 3712 a Black FD(93)p Black 1378 3658 a SDict begin H.R end 1378 3658 a 1378 3712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.93) cvn H.B /ANN pdfmark end 1378 3712 a FD(\))j(follo)n(ws)c(from)i(the)i(t)n(w)n(o)e (b)r(ounds)h(in)g(Equation)e(\()2916 3712 y SDict begin H.S end 2916 3712 a Black FD(92)p Black 2999 3658 a SDict begin H.R end 2999 3658 a 2999 3712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.92) cvn H.B /ANN pdfmark end 2999 3712 a FD(\),)j(Equation)523 3812 y(\()555 3812 y SDict begin H.S end 555 3812 a Black FD(94)p Black 638 3758 a SDict begin H.R end 638 3758 a 638 3812 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.93) cvn H.B /ANN pdfmark end 638 3812 a FD(\))28 b(is)f(the)h(statemen)n(t)e(of)i(the)g (Lemma.)648 3911 y(It)41 b(remains)e(to)i(pro)n(v)n(e)f(the)i (inequalities)37 b(in)j(\()2177 3911 y SDict begin H.S end 2177 3911 a Black FD(92)p Black 2261 3858 a SDict begin H.R end 2261 3858 a 2261 3911 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.92) cvn H.B /ANN pdfmark end 2261 3911 a FD(\).)h(W)-7 b(e)42 b(sho)n(w)f(b)r(elo)n(w)f(that)h(for)g(all) 523 4011 y(1)34 b Fs(\024)g Fv(n)g(<)g FD(1)p Fv(=\015)5 b FD(:)33 b Fv(h)1117 3981 y Fq(0)1140 4011 y FD(\()p Fv(n)p FD(\))h Fv(>)g FD(0,)g(and)g(that)h(for)f(all)e Fv(n)i(>)g(\014)t FD(\()p Fv(\015)5 b FD(\))p Fv(=\015)g FD(:)34 b Fv(h)2638 3981 y Fq(0)2661 4011 y FD(\()p Fv(n)p FD(\))h Fv(<)e FD(0,)h(this)g(pro)n(v)n(es)523 4110 y(Equation)26 b(\()914 4110 y SDict begin H.S end 914 4110 a Black FD(92)p Black 997 4057 a SDict begin H.R end 997 4057 a 997 4110 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.92) cvn H.B /ANN pdfmark end 997 4110 a FD(\).)648 4210 y(The)h(deriv)-5 b(ativ)n(e)25 b(of)j Fv(h)f FD(\(with)g(resp)r(ect)h(to)f Fv(n)p FD(\))h(is)1101 4429 y Fv(h)1149 4395 y Fq(0)1172 4429 y FD(\()p Fv(n)p FD(\))23 b(=)1407 4373 y(\(ln)14 b Fv(n)k FD(+)g(1\)\()p Fv(n)g FD(+)g(1)p Fv(=\015)5 b FD(\))2095 4343 y FC(2)2150 4373 y Fs(\000)18 b FD(2\()p Fv(n)g FD(+)g(1)p Fv(=\015)5 b FD(\))p Fv(n)14 b FD(ln)e Fv(n)p 1407 4410 1411 4 v 1920 4486 a FD(\()p Fv(n)19 b FD(+)f(1)p Fv(=\015)5 b FD(\))2268 4462 y FC(4)1309 4659 y FD(=)1407 4603 y(\(ln)14 b Fv(n)k FD(+)g(1\)\()p Fv(n)g FD(+)g(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g FD(2)p Fv(n)c FD(ln)e Fv(n)p 1407 4640 1027 4 v 1729 4716 a FD(\()p Fv(n)18 b FD(+)g(1)p Fv(=\015)5 b FD(\))2076 4692 y FC(3)1309 4888 y FD(=)1407 4832 y(\()p Fv(n)19 b FD(+)f(1)p Fv(=\015)5 b FD(\))17 b(+)h(\(1)p Fv(=\015)23 b Fs(\000)18 b Fv(n)p FD(\))c(ln)f Fv(n)p 1407 4869 942 4 v 1686 4945 a FD(\()p Fv(n)18 b FD(+)h(1)p Fv(=\015)5 b FD(\))2034 4921 y FC(3)2358 4888 y Fv(:)p Black 1922 5173 a FD(47)p Black eop end end %%Page: 48 48 TeXDict begin HPSdict begin 48 47 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.48) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(as)21 b(the)i(denominator)c(of)j Fv(h)1373 418 y Fq(0)1396 448 y FD(\()p Fv(n)p FD(\))h(is)e(p)r(ositiv) n(e,)e(the)k(nominator)c(determines)h(the)i(sign.)f(Ergo,)523 548 y(it)27 b(is)g(enough)g(to)g(lo)r(ok)f(at)h(the)h(sign)e(of)1485 695 y(1)p 1482 732 48 4 v 1482 808 a Fv(\015)1540 751 y Fs(\001)1563 684 y Fu(\000)1601 751 y Fv(\015)5 b(n)18 b FD(+)g(1)g(+)g(\(1)g Fs(\000)g Fv(\015)5 b(n)p FD(\))14 b(ln)f Fv(n)2395 684 y Fu(\001)2433 751 y Fv(;)523 964 y FD(and,)28 b(again,)d(as)i(1)p Fv(=\015)g(>)22 b FD(0,)27 b(this)g(is)g(equal)f(to)h(the)h(sign)f(of)1420 1132 y Fv(k)s FD(\()p Fv(n)p FD(\))d(=)e Fv(\015)5 b(n)18 b FD(+)g(1)g(+)g(\(1)g Fs(\000)g Fv(\015)5 b(n)p FD(\))14 b(ln)f Fv(n:)648 1301 y FD(F)-7 b(or)27 b(1)22 b Fs(\024)h Fv(n)g(<)f FD(1)p Fv(=\015)5 b FD(,)27 b(it)g(m)n(ust)f(b)r(e)i(that)g Fv(k)s FD(\()p Fv(n)p FD(\))c Fv(>)e FD(0,)27 b(and)h(hence)f Fv(h)2675 1270 y Fq(0)2699 1301 y FD(\()p Fv(n)p FD(\))c Fv(>)g FD(0.)648 1400 y(As)34 b Fv(\014)t FD(\()p Fv(\015)5 b FD(\))34 b Fv(>)g FD(1)g(and)g Fv(n)f Fs(\025)h Fv(\014)t FD(\()p Fv(\015)5 b FD(\))p Fv(=\015)39 b FD(w)n(e)34 b(ha)n(v)n(e)f(that)h Fv(n)g(>)f FD(1)p Fv(=\015)38 b FD(and)c Fv(n\015)39 b Fs(\025)33 b Fv(\014)t FD(\()p Fv(\015)5 b FD(\))35 b Fv(>)e FD(1.)523 1500 y(No)n(w,)27 b(w)n(e)g(ha)n(v)n(e)g(that)1039 1668 y Fv(k)s FD(\()p Fv(n)p FD(\))d(=)e Fv(\015)5 b(n)18 b FD(+)g(1)g(+)g(\(1)h Fs(\000)f Fv(\015)5 b(n)p FD(\))14 b(ln)n(\()p Fv(n)p FD(\))1223 1843 y(=)22 b(\()p Fv(\015)5 b(n)19 b Fs(\000)f FD(1\))1616 1726 y Fu(\022)1676 1843 y FD(1)g(+)1929 1787 y(2)p 1829 1824 241 4 v 1829 1900 a Fv(\015)5 b(n)18 b Fs(\000)g FD(1)2098 1843 y Fs(\000)g FD(ln)13 b Fv(n)2314 1726 y Fu(\023)1223 2075 y Fv(<)22 b FD(\()p Fv(\015)5 b(n)19 b Fs(\000)f FD(1\))1616 1958 y Fu(\022)1676 2075 y FD(1)g(+)1962 2019 y(2)p 1829 2056 307 4 v 1829 2132 a Fv(\014)t FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(1)2164 2075 y Fs(\000)g FD(ln)13 b(\(1)p Fv(=\015)5 b FD(\))2526 1958 y Fu(\023)1223 2297 y FD(=)1353 2241 y Fv(\015)g(n)18 b Fs(\000)g FD(1)p 1320 2278 V 1320 2354 a Fv(\014)t FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(1)1637 2230 y Fu(\000)1675 2297 y FD(2)g Fs(\000)g FD(\()p Fv(\014)t FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(1\)\(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g FD(1\))2675 2230 y Fu(\001)1223 2528 y FD(=)1353 2472 y Fv(\015)5 b(n)18 b Fs(\000)g FD(1)p 1320 2509 V 1320 2585 a Fv(\014)t FD(\()p Fv(\015)5 b FD(\))19 b Fs(\000)f FD(1)1637 2411 y Fu(\022)1698 2528 y FD(2)g Fs(\000)2041 2472 y FD(2)p 1851 2509 422 4 v 1851 2585 a(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g FD(1)2300 2528 y Fs(\001)h FD(\(ln)13 b(\(1)p Fv(=\015)5 b FD(\))18 b Fs(\000)g FD(1)o(\))2827 2411 y Fu(\023)1223 2694 y FD(=)k(0)p Fv(:)648 2862 y FD(th)n(us)27 b(concluding)e(the)j(pro)r(of)f(of)h(\() 1732 2862 y SDict begin H.S end 1732 2862 a Black FD(92)p Black 1815 2809 a SDict begin H.R end 1815 2809 a 1815 2862 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.3.92) cvn H.B /ANN pdfmark end 1815 2862 a FD(\))g(and)g(the)g(lemma.)3350 2962 y Fs(u)-55 b(t)523 3100 y SDict begin H.S end 523 3100 a 523 3100 a SDict begin 12 H.A end 523 3100 a 523 3100 a SDict begin [/View [/XYZ H.V]/Dest (section.A.4) cvn /DEST pdfmark end 523 3100 a 116 x Fx(D)112 b(Exp)s(ected)49 b(size)g(of)g(the)g(Maxim)m(um)i(capacit)m(y)e Fe(d)721 3333 y(\015)6 b Fx(-Indep)s(enden)m(t)39 b(Set)f(\()p Fe(d)e Fx(iden)m(tical)i(mac)m(hines\))523 3520 y FD(A)29 b(capacit)n(y)d Fv(d)i(\015)5 b FD(-indep)r(enden)n(t)28 b(set)g(is)f(a)h(set)h(of)f(\\feasible)d(ren)n(tals",)h(giv)n(en)h (that)h Fv(d)h FD(items)523 3620 y(can)19 b(b)r(e)g(ren)n(ted,)g(and)f (eac)n(h)g(item)g(is)g(ren)n(ted)g(for)h(a)f(p)r(erio)r(d)g(of)h (length)f Fv(\015)5 b FD(.)19 b(Equiv)-5 b(alen)n(tly)15 b(to)k(the)523 3720 y(de\014nition)g(in)g(Section)1258 3720 y SDict begin H.S end 1258 3720 a Black FD(2)p Black 1299 3666 a SDict begin H.R end 1299 3666 a 1299 3720 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.1.2) cvn H.B /ANN pdfmark end 1299 3720 a FD(,)i(a)f(capacit)n(y)e Fv(d)j(\015)5 b FD(-indep)r(enden)n(t)19 b(set)i(is)e(a)h(set)h(of)f(p) r(oin)n(ts)f Fv(S)28 b Fs(\032)23 b FD([0)p Fv(;)14 b FD(1\),)523 3819 y(suc)n(h)32 b(that)g(giv)n(en)e(an)n(y)i(subset)g(of) g(in)n(terv)-5 b(als)29 b Fv(I)7 b FD(,)33 b Fs(j)p Fv(I)7 b Fs(j)30 b Fv(>)g(d)p FD(,)j Fv(I)k Fs(\032)31 b(f)p FD([)p Fv(t;)14 b(t)21 b FD(+)g Fv(\015)5 b FD(\))30 b Fs(j)h Fv(t)f Fs(2)h Fv(S)5 b Fs(g)p FD(,)32 b(the)523 3919 y(in)n(tersection)27 b(of)j(all)e(in)n(terv)-5 b(als)27 b(in)j Fv(I)37 b FD(is)29 b(empt)n(y)-7 b(.)29 b(The)h(unit)g(in)n (terv)-5 b(al)27 b(graph)i(is)g(de\014ned)h(b)n(y)523 4019 y(in)n(terv)-5 b(als)25 b(of)i(length)g Fv(\015)32 b FD(whose)27 b(left)h(endp)r(oin)n(ts)e(are)h(the)h(p)r(oin)n(ts)e(of) i Fv(Z)6 b FD(.)648 4118 y(Let)25 b Fv(m)867 4130 y Ft(d)905 4118 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))26 b(denote)f(the)g(size)f (of)h(a)g(maxim)n(um)20 b(capacit)n(y)j Fv(d)j(\015)5 b FD(-indep)r(enden)n(t)24 b(subset)523 4218 y(of)h(a)h(set)f Fv(Z)6 b FD(.)25 b(In)h(this)f(section)f(w)n(e)h(study)h(the)g(exp)r (ectation)e(of)h Fv(m)2556 4230 y Ft(d)2595 4218 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))26 b(when)f(the)h(p)r(oin)n(ts)523 4317 y(of)i Fv(Z)33 b FD(are)26 b(c)n(hosen)h(uniformly)e(at)i(random)f (in)h([0)p Fv(;)14 b FD(1\).)27 b(Sp)r(eci\014cally)d(w)n(e)j(pro)n(v)n (e:)523 4338 y SDict begin H.S end 523 4338 a 523 4338 a SDict begin 12 H.A end 523 4338 a 523 4338 a SDict begin [/View [/XYZ H.V]/Dest (theorem.A.4.5) cvn /DEST pdfmark end 523 4338 a Black 140 x Fr(Theorem)j(5.)p Black 41 w Fp(L)l(et)j Fv(Z)39 b Fp(b)l(e)33 b(a)h(set)f(of)h Fv(n)f Fp(p)l(oints)g(chosen)i(uniformly)f(at)f(r)l(andom)h(in)f FD([0)p Fv(;)14 b FD(1\))p Fp(,)523 4577 y(and)30 b(let)g Fv(\015)e FD(=)22 b(1)p Fv(=k)s Fp(,)29 b(for)i(some)f(inte)l(ger)g Fv(k)c Fs(\025)c FD(2)p Fp(.)30 b(Then)g(we)g(have)1004 4821 y SDict begin H.S end 1004 4821 a 1004 4821 a SDict begin 12 H.A end 1004 4821 a 1004 4821 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.95) cvn /DEST pdfmark end 1004 4821 a FD(E)1061 4833 y Ft(Z)1114 4821 y FD([)p Fv(m)1210 4833 y Ft(d)1248 4821 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])24 b Fs(\025)e FD(min)12 b(\()p Fv(n;)28 b(d)19 b Fs(\001)g FD(1)p Fv(=\015)5 b FD(\))17 b Fs(\001)2200 4679 y Fu( )2266 4821 y FD(1)h Fs(\000)g Fv(\002)2474 4679 y Fu( )2540 4688 y(r)p 2623 4688 147 4 v 2633 4764 a FD(ln)13 b Fv(d)p 2633 4801 127 4 v 2675 4878 a(d)2770 4679 y Fu(!)o(!)2901 4821 y Fv(:)333 b FD(\(95\))p Black 1922 5173 a(48)p Black eop end end %%Page: 49 49 TeXDict begin HPSdict begin 49 48 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.49) cvn /DEST pdfmark end 515 125 a Black Black Black 523 448 a Fp(Pr)l(o)l(of.)p Black 43 w FD(Let)28 b Fv(Y)46 b FD(b)r(e)28 b(a)f(random)f(subset)i(of)f Fv(Z)34 b FD(of)27 b(size)f(min)13 b(\()p Fv(n;)h(T)e FD(\))27 b(where)523 548 y Fv(T)34 b FD(=)23 b(\()p Fv(d)15 b Fs(\000)864 477 y(p)p 933 477 225 4 v 71 x FD(3)p Fv(d)f FD(ln)e Fv(d)p FD(\))k Fs(\001)e FD(1)p Fv(=\015)19 b FD(+)14 b(1.)26 b(Since)f(for)g(an)n(y)g(subset)h Fv(Y)42 b Fs(\022)22 b Fv(Z)6 b FD(,)26 b Fv(m)2668 560 y Ft(d)2706 548 y FD(\()p Fv(Y)5 b(;)14 b(\015)5 b FD(\))23 b Fs(\024)g Fv(m)3092 560 y Ft(d)3130 548 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))26 b(a)523 648 y(lo)n(w)n(er)h(b)r(ound)j(on)g Fv(m)1191 660 y Ft(d)1230 648 y FD(\()p Fv(Y)5 b(;)14 b(\015)5 b FD(\))29 b(is)g(also)f(a)h(lo)n(w)n(er)e(b)r(ound)j(on)g Fv(m)2455 660 y Ft(d)2494 648 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\).)30 b(As)f Fv(Z)36 b FD(is)29 b(a)g(random)523 747 y(set)d(of)g(p)r(oin)n(ts)f(and)h Fv(Y)44 b FD(is)25 b(a)h(random)e(subset)i(of)g Fv(Z)6 b FD(,)25 b(c)n(ho)r(osing)f Fv(Z)31 b FD(\014rst)26 b(and)g(then)g(c)n(ho)r(osing)523 847 y Fv(Y)46 b FD(giv)n(es)26 b(the)i(same)e(distribution)e(on)k Fv(Y)46 b FD(as)27 b(do)r(es)g(simply)e(c)n(ho)r(osing)g Fv(Y)46 b FD(at)28 b(random.)648 952 y(Let)i Fv(Y)866 922 y Fq(0)917 952 y FD(=)d Fs(f)p Fv(x)h Fs(2)g Fv(Y)46 b Fs(j)28 b(j)p FD([)p Fv(x)19 b Fs(\000)f Fv(\015)5 b(;)14 b(x)p FD(\))19 b Fs(\\)g Fv(Y)f Fs(j)28 b Fv(<)f(d)p Fs(g)k FD(b)r(e)f(the)h(subset)g(of)f Fv(Y)49 b FD(con)n(taining)27 b(only)523 1052 y(p)r(oin)n(ts)33 b Fv(x)i FD(that)f(do)g(not)g(ha)n(v) n(e)f Fv(d)i FD(p)r(oin)n(ts)e(in)h(an)f(in)n(terv)-5 b(al)32 b(of)i(length)f Fv(\015)39 b FD(ending)33 b(at)i Fv(x)p FD(.)f(So)523 1151 y Fv(Y)590 1121 y Fq(0)639 1151 y FD(is)25 b(a)h(capacit)n(y)e Fv(d)i(\015)5 b FD(-indep)r(enden)n (t)25 b(set)h(\(as)g(the)g(greedy)f(algorithm)d(applied)i(to)i Fv(Y)3226 1121 y Fq(0)3276 1151 y FD(will)523 1251 y(pic)n(k)g(all)g (the)i(p)r(oin)n(ts\).)e(Therefore)h(for)g(an)n(y)g(\014xed)g Fv(Z)6 b FD(,)1392 1445 y SDict begin H.S end 1392 1445 a 1392 1445 a SDict begin 12 H.A end 1392 1445 a 1392 1445 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.96) cvn /DEST pdfmark end 1392 1445 a FD(E)1449 1457 y Ft(Y)14 b Fq(\022)p Ft(Z)1608 1445 y FD([)p Fv(m)1704 1457 y Ft(d)1742 1445 y FD(\()p Fv(Y)1841 1410 y Fq(0)1865 1445 y Fv(;)g(\015)5 b FD(\)])22 b(=)h(E)2172 1457 y Ft(Y)14 b Fq(\022)p Ft(Z)2330 1445 y FD([)p Fs(j)p Fv(Y)2443 1410 y Fq(0)2466 1445 y Fs(j)p FD(])p Fv(:)722 b FD(\(96\))523 1638 y(Since)27 b Fv(Y)806 1608 y Fq(0)853 1638 y Fs(\022)22 b Fv(Y)42 b Fs(\022)23 b Fv(Z)33 b FD(w)n(e)27 b(ha)n(v)n(e)866 1832 y SDict begin H.S end 866 1832 a 866 1832 a SDict begin 12 H.A end 866 1832 a 866 1832 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.97) cvn /DEST pdfmark end 866 1832 a FD(E)923 1844 y Ft(Z)976 1832 y FD([)q(E)1057 1844 y Ft(Y)13 b Fq(\022)p Ft(Z)1215 1832 y FD([)p Fv(m)1311 1844 y Ft(d)1349 1832 y FD(\()p Fv(Y)1448 1798 y Fq(0)1472 1832 y Fv(;)h(\015)5 b FD(\)])o(])24 b Fs(\024)e FD(E)1802 1844 y Ft(Z)1855 1832 y FD([E)1935 1844 y Ft(Y)14 b Fq(\022)p Ft(Z)2093 1832 y FD([)p Fv(m)2189 1844 y Ft(d)2228 1832 y FD(\()p Fv(Y)5 b(;)14 b(\015)5 b FD(\)]])23 b Fs(\024)g FD(E)2644 1844 y Ft(Z)2697 1832 y FD([)p Fv(m)2793 1844 y Ft(d)2831 1832 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])196 b(\(97\))523 2026 y(F)-7 b(rom)17 b(\()764 2026 y SDict begin H.S end 764 2026 a Black FD(96)p Black 847 1972 a SDict begin H.R end 847 1972 a 847 2026 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.96) cvn H.B /ANN pdfmark end 847 2026 a FD(\))i(and)f(\()1082 2026 y SDict begin H.S end 1082 2026 a Black FD(97)p Black 1166 1972 a SDict begin H.R end 1166 1972 a 1166 2026 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.97) cvn H.B /ANN pdfmark end 1166 2026 a FD(\))g(follo)n(ws)e(that)i(it)g(su\016ces)g(to)g(sho)n (w)g(a)g(lo)n(w)n(er)e(b)r(ound)i(on)g(E)2963 2038 y Ft(Z)3016 2026 y FD([E)3096 2038 y Ft(Y)c Fq(\022)p Ft(Z)3255 2026 y FD([)p Fs(j)p Fv(Y)3367 1996 y Fq(0)3391 2026 y Fs(j)p FD(]].)695 2220 y SDict begin H.S end 695 2220 a 695 2220 a SDict begin 12 H.A end 695 2220 a 695 2220 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.98) cvn /DEST pdfmark end 695 2220 a FD(E)752 2232 y Ft(Z)805 2220 y FD([E)885 2232 y Ft(Y)g Fq(\022)p Ft(Z)1044 2220 y FD([)p Fs(j)p Fv(Y)1156 2185 y Fq(0)1180 2220 y Fs(j)p FD(]])719 2363 y(=)813 2284 y Fu(X)806 2462 y Ft(y)r Fq(2)p Ft(Y)954 2363 y FD(1)k Fs(\001)h FD(Prob)n([)14 b Fs(j)p FD([)p Fv(y)21 b Fs(\000)d Fv(\015)5 b(;)14 b(y)s FD(\))k Fs(\\)h Fv(Y)g Fs(j)k Fv(<)g(d)14 b FD(])719 2578 y Fs(\025)22 b(j)q Fv(Y)c Fs(j)h(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fs(j)p Fv(Y)k Fs(j)h(\000)f FD(1)p Fv(;)c(\015)5 b FD(\))22 b Fs(\024)h Fv(d)18 b Fs(\000)g FD(1)c(])719 2702 y(=)22 b Fs(j)q Fv(Y)c Fs(j)h(\001)f FD(\(1)g Fs(\000)g FD(Prob)o([)c Fv(B)t(in)p FD(\(min)e(\()p Fv(n;)i(T)e FD(\))18 b Fs(\000)g FD(1)o Fv(;)c(\015)5 b FD(\))23 b Fs(\025)g Fv(d)14 b FD(]\))719 2827 y Fs(\025)22 b(j)q Fv(Y)c Fs(j)h(\001)f FD(\(1)g Fs(\000)g FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(T)29 b Fs(\000)18 b FD(1\))p Fv(;)c(\015)5 b FD(\))23 b Fs(\025)g Fv(d)14 b FD(]\))719 3001 y(=)22 b Fs(j)q Fv(Y)c Fs(j)h(\001)979 2884 y Fu(\022)1040 3001 y FD(1)f Fs(\000)g FD(Prob)1360 2884 y Fu(\024)1418 3001 y Fv(B)t(in)p FD(\()p Fv(T)29 b Fs(\000)18 b FD(1)p Fv(;)c(\015)5 b FD(\))22 b Fs(\025)2026 2884 y Fu(\022)2087 3001 y FD(1)c(+)2240 2945 y Fv(d)h Fs(\000)f FD(\()p Fv(T)29 b Fs(\000)18 b FD(1\))p Fv(\015)p 2240 2982 460 4 v 2312 3058 a FD(\()p Fv(T)30 b Fs(\000)18 b FD(1\))p Fv(\015)2710 2884 y Fu(\023)2771 3001 y FD(\()p Fv(T)30 b Fs(\000)18 b FD(1\))p Fv(\015)3100 2884 y Fu(\025\023)719 3258 y FD(=)k Fs(j)q Fv(Y)c Fs(j)h(\001)979 3116 y Fu( )1045 3258 y FD(1)f Fs(\000)g FD(Prob)1365 3116 y Fu(")1427 3258 y Fv(B)t(in)p FD(\()p Fv(T)29 b Fs(\000)18 b FD(1)p Fv(;)c(\015)5 b FD(\))23 b Fs(\025)2035 3116 y Fu( )2101 3258 y FD(1)18 b(+)2264 3131 y Fs(p)p 2334 3131 225 4 v 2334 3202 a FD(3)p Fv(d)c FD(ln)e Fv(d)p 2254 3239 316 4 v 2254 3315 a FD(\()p Fv(T)30 b Fs(\000)18 b FD(1\))p Fv(\015)2579 3116 y Fu(!)2645 3258 y FD(\()p Fv(T)30 b Fs(\000)18 b FD(1\))p Fv(\015)2974 3116 y Fu(#!)3116 3258 y Fv(:)p Black 118 w FD(\(98\))p Black 523 3527 a(The)36 b(deriv)-5 b(ation)33 b(follo)n(wing)f(the)37 b(4th)e(line)g(ab)r(o)n(v)n(e)f(follo)n(ws)f(since)i Fv(d)i FD(=)g(\(1)23 b(+)h(\()p Fv(d)g Fs(\000)g FD(\()p Fv(T)35 b Fs(\000)523 3626 y FD(1\))p Fv(\015)5 b FD(\))p Fv(=)p FD(\()p Fv(T)22 b Fs(\000)12 b FD(1\))p Fv(\015)5 b FD(\)\()p Fv(T)22 b Fs(\000)12 b FD(1\))p Fv(\015)5 b FD(.)23 b(Applying)f(the)j(m)n(ultiplicativ)n(e)18 b(form)23 b(of)h(the)h(Cherno\013)f(b)r(ound)523 3756 y(\(Prob)o([)14 b Fv(X)29 b Fs(\025)23 b FD(\(1)18 b(+)g Fv(\017)p FD(\))p Fv(\026)c FD(])34 b Fs(\024)g Fv(e)1455 3726 y Fq(\000)1516 3699 y Fo(\017)1541 3682 y Fb(2)1574 3699 y Fo(\026)p 1516 3713 94 3 v 1549 3746 a Fb(3)1658 3756 y FD(when)g Fv(X)41 b FD(is)33 b(sum)h(of)g Fv(n)g FD(I)r(ID)h(random)d(v)-5 b(ariables)31 b(and)523 3873 y Fv(\026)23 b FD(=)g(E[)o Fv(X)7 b FD(])27 b(on)h Fv(B)t(in)p FD(\()p Fv(T)h Fs(\000)18 b FD(1)p Fv(;)c(\015)5 b FD(\))27 b(with)g Fv(\017)c FD(=)1873 3791 y Fq(p)p 1928 3791 181 3 v 49 x FC(3)p Ft(d)11 b FC(ln)g Ft(d)p 1873 3854 235 4 v 1879 3902 a FC(\()p Ft(T)e Fq(\000)p FC(1\))p Ft(\015)2146 3873 y FD(w)n(e)27 b(obtain)1120 4201 y(E)1177 4213 y Ft(Y)14 b Fq(\022)p Ft(Z)1335 4201 y FD([)p Fs(j)p Fv(Y)1448 4167 y Fq(0)1471 4201 y Fs(j)p FD(])23 b Fs(\025)g(j)p Fv(Y)c Fs(j)f(\001)1801 4084 y Fu(\022)1862 4201 y FD(1)g Fs(\000)g Fv(e)2044 4154 y Fq(\000)2106 4132 y Fb(1)p 2106 4141 29 3 v 2106 4174 a(3)2144 4154 y Fq(\001)2164 4090 y Fa(\020)2214 4096 y Fd(p)p 2259 4096 158 3 v 2259 4132 a Fb(3)p Fo(d)10 b Fb(ln)g Fo(d)p 2214 4142 204 3 v 2219 4175 a Fb(\()p Fo(T)d Fd(\000)p Fb(1\))p Fo(\015)2427 4090 y Fa(\021)2467 4108 y Fb(2)2500 4154 y Fq(\001)p FC(\()p Ft(T)i Fq(\000)p FC(1\))p Ft(\015)2747 4084 y Fu(\023)1540 4408 y FD(=)23 b Fs(j)p Fv(Y)c Fs(j)f(\001)1801 4316 y Fu(\020)1850 4408 y FD(1)g Fs(\000)g Fv(e)2032 4368 y Fq(\000)2126 4346 y Fo(d)10 b Fb(ln)g Fo(d)p 2094 4355 194 3 v 2094 4389 a Fb(\()p Fo(T)d Fd(\000)p Fb(1\))p Fo(\015)2302 4316 y Fu(\021)1540 4592 y FD(=)23 b Fs(j)p Fv(Y)c Fs(j)f(\001)1801 4500 y Fu(\020)1850 4592 y FD(1)g Fs(\000)g Fv(e)2032 4547 y Fq(\000)2169 4524 y Fo(d)10 b Fb(ln)g Fo(d)p 2094 4533 280 3 v 2094 4576 a(d)p Fd(\000)2170 4541 y(p)p 2215 4541 158 3 v 2215 4576 a Fb(3)p Fo(d)g Fb(ln)g Fo(d)2387 4500 y Fu(\021)1540 4775 y Fs(\025)23 b(j)p Fv(Y)c Fs(j)f(\001)1801 4683 y Fu(\020)1850 4775 y FD(1)g Fs(\000)g Fv(e)2032 4741 y Fq(\000)2094 4718 y Fo(d)10 b Fb(ln)g Fo(d)p 2094 4727 130 3 v 2143 4761 a(d)2238 4683 y Fu(\021)1540 4924 y FD(=)23 b Fs(j)p Fv(Y)c Fs(j)f(\001)h FD(\(1)f Fs(\000)g FD(1)p Fv(=d)p FD(\))27 b Fv(:)p Black 1922 5173 a FD(49)p Black eop end end %%Page: 50 50 TeXDict begin HPSdict begin 50 49 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.50) cvn /DEST pdfmark end 515 125 a Black Black 523 461 a FD(Substituting)27 b Fs(j)p Fv(Y)18 b Fs(j)23 b FD(=)g(min)1374 369 y Fu(\020)1424 461 y Fv(n;)14 b FD(\()p Fv(d)19 b Fs(\000)1688 390 y(p)p 1757 390 225 4 v 71 x FD(3)p Fv(d)14 b FD(ln)e Fv(d)q FD(\))18 b Fs(\001)h FD(1)p Fv(=\015)j FD(+)c(1)2348 369 y Fu(\021)2425 461 y FD(w)n(e)27 b(get)975 728 y(E)1032 740 y Ft(Y)14 b Fq(\022)p Ft(Z)1190 728 y FD([)p Fs(j)p Fv(Y)1303 694 y Fq(0)1326 728 y Fs(j)p FD(])24 b Fs(\025)f FD(min)1636 636 y Fu(\020)1685 728 y Fv(n;)1772 636 y Fu(\020)1822 728 y Fv(d)18 b Fs(\000)1966 653 y(p)p 2036 653 V 2036 728 a FD(3)p Fv(d)c FD(ln)e Fv(d)2260 636 y Fu(\021)2328 728 y Fs(\001)19 b FD(1)p Fv(=\015)2502 636 y Fu(\021)2569 728 y Fs(\001)f FD(\(1)g Fs(\000)g FD(1)p Fv(=d)p FD(\))1396 961 y Fs(\025)23 b FD(min)1636 868 y Fu(\020)1685 961 y Fv(n;)14 b(d=\015)1905 868 y Fu(\021)1972 961 y Fs(\001)2014 819 y Fu( )2080 961 y FD(1)k Fs(\000)2233 834 y(p)p 2302 834 182 4 v 70 x FD(3)c(ln)e Fv(d)p 2233 941 251 4 v 2302 958 a Fs(p)p 2371 958 44 4 v 71 x Fv(d)2493 819 y Fu(!)2578 961 y Fs(\001)18 b FD(\(1)g Fs(\000)g FD(1)p Fv(=d)p FD(\))1396 1243 y Fs(\025)23 b FD(min)1636 1151 y Fu(\020)1685 1243 y Fv(n;)14 b(d=\015)1905 1151 y Fu(\021)1972 1243 y Fs(\001)2014 1101 y Fu( )2080 1243 y FD(1)k Fs(\000)2233 1116 y(p)p 2302 1116 182 4 v 71 x FD(3)c(ln)e Fv(d)p 2233 1224 251 4 v 2302 1240 a Fs(p)p 2371 1240 44 4 v 71 x Fv(d)2512 1243 y Fs(\000)18 b FD(1)p Fv(=d)2722 1101 y Fu(!)1396 1525 y FD(=)23 b(min)1636 1433 y Fu(\020)1685 1525 y Fv(n;)14 b(d=\015)1905 1433 y Fu(\021)1972 1525 y Fs(\001)2014 1383 y Fu( )2080 1525 y FD(1)k Fs(\000)g Fv(\002)2288 1383 y Fu( )2354 1393 y(r)p 2437 1393 147 4 v 2447 1469 a FD(ln)13 b Fv(d)p 2447 1506 127 4 v 2489 1582 a(d)2583 1383 y Fu(!!)2742 1525 y Fv(:)3350 1793 y Fs(u)-55 b(t)523 1907 y SDict begin H.S end 523 1907 a 523 1907 a SDict begin 12 H.A end 523 1907 a 523 1907 a SDict begin [/View [/XYZ H.V]/Dest (theorem.A.4.6) cvn /DEST pdfmark end 523 1907 a Black 113 x Fr(Theorem)30 b(6.)p Black 41 w Fp(L)l(et)i Fv(Z)37 b Fp(b)l(e)32 b(a)g(set)g(of)g Fv(n)27 b FD(=)f Fv(d)21 b Fs(\001)f FD(1)p Fv(=\015)35 b Fp(p)l(oints)d(chosen)h(uniformly)f(at)g(r)l(andom)523 2120 y(in)e FD([0)p Fv(;)14 b FD(1\))29 b Fp(and)h(let)g Fv(\015)d FD(=)c(1)p Fv(=k)32 b Fp(\(for)e(some)g(inte)l(ger)g Fv(k)c Fs(\025)d FD(2)p Fp(\).)29 b(Then)i(we)f(have)1312 2379 y SDict begin H.S end 1312 2379 a 1312 2379 a SDict begin 12 H.A end 1312 2379 a 1312 2379 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.99) cvn /DEST pdfmark end 1312 2379 a FD(E)1369 2391 y Ft(Z)1422 2379 y FD([)p Fv(m)1518 2391 y Ft(d)1557 2379 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\024)g Fv(n)18 b Fs(\001)2007 2262 y Fu(\022)2068 2379 y FD(1)g Fs(\000)g Fv(\002)2276 2262 y Fu(\022)2384 2323 y FD(1)p 2348 2360 113 4 v 2348 2376 a Fs(p)p 2417 2376 44 4 v 71 x Fv(d)2471 2262 y Fu(\023\023)2593 2379 y Fv(:)641 b FD(\(99\))p Black 523 2651 a Fp(Pr)l(o)l(of.)p Black 43 w FD(F)-7 b(or)29 b(an)n(y)g(0)c Fs(\024)h Fv(i)g(<)f FD(1)p Fv(=\015)33 b FD(de\014ne)d Fv(Z)1855 2663 y Ft(i)1908 2651 y Fs(\022)c Fv(Z)35 b FD(to)30 b(b)r(e)f(the)h(set)g Fs(f)p Fv(z)f Fs(2)d Fv(Z)32 b Fs(j)26 b Fv(z)j Fs(2)e FD([)p Fv(i\015)33 b(;)44 b FD(\()p Fv(i)19 b FD(+)523 2751 y(1\))p Fv(\015)5 b FD(\))p Fs(g)p FD(,)27 b(all)e(the)j(p)r(oin)n(ts)f(arriving)d(at)j (the)h Fv(i)p FD(-th)f(slice)f(of)h(size)g Fv(\015)5 b FD(.)648 2864 y(Ob)n(viously)1038 2801 y Fu(S)1107 2889 y FC(1)p Fq(\024)p Ft(i<)p FC(1)p Ft(=\015)1391 2864 y Fv(Z)1448 2876 y Ft(i)1498 2864 y FD(=)23 b Fv(Z)6 b FD(.)27 b(Therefore)1510 3184 y SDict begin H.S end 1510 3184 a 1510 3184 a SDict begin 12 H.A end 1510 3184 a 1510 3184 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.100) cvn /DEST pdfmark end 1510 3184 a Fv(m)1583 3196 y Ft(d)1622 3184 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\))23 b Fs(\024)1946 3073 y FC(1)p Ft(=\015)1939 3105 y Fu(X)1945 3282 y Ft(i)p FC(=0)2073 3184 y Fv(m)2146 3196 y Ft(d)2185 3184 y FD(\()p Fv(Z)2274 3196 y Ft(i)2301 3184 y Fv(;)14 b(\015)5 b FD(\))798 b(\(100\))523 3470 y(T)-7 b(o)22 b(giv)n(e)f(un)i(upp)r(er)g(b)r(ound)h(on)e(the)h(size)f(of)h Fv(m)1970 3482 y Ft(d)2008 3470 y FD(\()p Fv(Z)2097 3482 y Ft(i)2125 3470 y Fv(;)14 b(\015)5 b FD(\))22 b(notice)g(that)h(if)f Fs(j)p Fv(Z)2830 3482 y Ft(i)2858 3470 y Fs(j)h(\024)g Fv(d)g FD(then)g Fv(Z)3299 3482 y Ft(i)3349 3470 y FD(is)523 3570 y(a)k(capacit)n(y)f Fv(d)i(\015)5 b FD(-indep)r(enden)n(t)27 b(set,)h(th)n(us)g Fv(m)1946 3582 y Ft(d)1984 3570 y FD(\()p Fv(Z)2073 3582 y Ft(i)2101 3570 y Fv(;)14 b(\015)5 b FD(\))23 b(=)g Fs(j)p Fv(Z)2409 3582 y Ft(i)2436 3570 y Fs(j)p FD(.)28 b(Otherwise,)e(an)n(y)h(subset)h Fv(B)523 3669 y FD(of)23 b Fv(Z)670 3681 y Ft(i)720 3669 y FD(is)f(capacit)n(y)e Fv(d)j(\015)5 b FD(-indep)r(enden)n(t)22 b(set)h(i\013)f Fs(j)p Fv(B)t Fs(j)i(\024)e Fv(d)p FD(.)h(Th)n(us)g Fv(m)2539 3681 y Ft(d)2577 3669 y FD(\()p Fv(Z)2666 3681 y Ft(i)2694 3669 y Fv(;)14 b(\015)5 b FD(\))23 b(=)f Fv(d)p FD(.)i(Com)n(bining)523 3769 y(these)k(observ)-5 b(ations)25 b(w)n(e)i(get)1341 3978 y SDict begin H.S end 1341 3978 a 1341 3978 a SDict begin 12 H.A end 1341 3978 a 1341 3978 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.101) cvn /DEST pdfmark end 1341 3978 a FD(E)1398 3990 y Ft(Z)1451 3978 y FD([)p Fv(m)1547 3990 y Ft(d)1586 3978 y FD(\()p Fv(Z)1675 3990 y Ft(i)1703 3978 y Fv(;)14 b(\015)5 b FD(\)])23 b(=)f(E)2010 3990 y Ft(Z)2063 3978 y FD([)p Fv(min)p FD(\()p Fs(j)p Fv(Z)2350 3990 y Ft(i)2377 3978 y Fs(j)p Fv(;)14 b(d)p FD(\)])28 b Fv(:)630 b FD(\(101\))523 4187 y Fs(j)p Fv(Z)603 4199 y Ft(i)630 4187 y Fs(j)24 b FD(coun)n(t)f(the)h(n)n(um)n(b)r(er)e (of)h(p)r(oin)n(ts)f(that)i(fall)d(in)n(to)h(the)i(line)e(segmen)n(t)f ([)p Fv(i\015)5 b(;)14 b FD(\()p Fv(i)c FD(+)g(1\))p Fv(\015)5 b FD(\))22 b(when)523 4287 y(w)n(e)i(thro)n(w)f Fv(n)g FD(=)f Fv(d=\015)29 b FD(p)r(oin)n(ts)23 b(uniformly)e(at)i (random)g(in)n(to)f(the)j(line)d(segmen)n(t)g([0)p Fv(;)14 b FD(1\).)24 b(Ergo,)523 4387 y Fs(j)p Fv(Z)603 4399 y Ft(i)630 4387 y Fs(j)k FD(is)f(distributed)f(as)h Fv(B)t(in)p FD(\()o Fv(d)19 b Fs(\001)g FD(1)p Fv(=\015)5 b(;)26 b(\015)5 b FD(\),)28 b(therefore)1183 4596 y(E)1240 4608 y Ft(Z)1293 4596 y FD([)p Fv(m)1389 4608 y Ft(d)1427 4596 y FD(\()p Fv(Z)1516 4608 y Ft(i)1544 4596 y Fv(;)14 b(\015)5 b FD(\)])23 b(=)f(E[min)n(\()p Fv(B)t(in)p FD(\()p Fv(d)d Fs(\001)g FD(1)p Fv(=\015)5 b(;)26 b(\015)5 b FD(\))p Fv(;)14 b(d)p FD(\)])p Fv(:)523 4805 y FD(F)-7 b(rom)17 b(Lemma)f(\()1051 4805 y SDict begin H.S end 1051 4805 a Black FD(28)p Black 1134 4751 a SDict begin H.R end 1134 4751 a 1134 4805 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (lemma.28) cvn H.B /ANN pdfmark end 1134 4805 a FD(\))j(\(see)f(b)r(elo)n(w\))g(and)g(the)h(equation)d(ab)r(o)n(v)n(e)h (w)n(e)h(conclude)g(that)g(E)3115 4817 y Ft(Z)3168 4805 y FD([)p Fv(m)3264 4817 y Ft(d)3303 4805 y FD(\()p Fv(Z)3392 4817 y Ft(i)3419 4805 y Fv(;)c(\015)5 b FD(\)])23 b Fs(\024)523 4921 y Fv(d)566 4853 y Fu(\000)604 4921 y FD(1)e Fs(\000)763 4840 y Fq(p)p 817 4840 75 3 v 817 4888 a FC(2)p Ft(\031)p 763 4902 129 4 v 795 4949 a(e)826 4933 y Fb(2)923 4921 y Fs(\001)977 4840 y Fq(p)p 1032 4840 124 3 v 42 x FC(1)p Fq(\000)p Ft(\015)p 977 4902 178 4 v 1021 4911 a Fq(p)p 1076 4911 35 3 v 49 x Ft(d)1165 4853 y Fu(\001)1203 4921 y FD(.)32 b(Substituting)f(this)g(b)r(ound)h(in)n(to)f(\()2369 4921 y SDict begin H.S end 2369 4921 a Black FD(101)p Black 2494 4867 a SDict begin H.R end 2494 4867 a 2494 4921 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.101) cvn H.B /ANN pdfmark end 2494 4921 a FD(\))h(and)f(then)i(in)n(to)d(\()3121 4921 y SDict begin H.S end 3121 4921 a Black FD(100)p Black 3246 4867 a SDict begin H.R end 3246 4867 a 3246 4921 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.100) cvn H.B /ANN pdfmark end 3246 4921 a FD(\))i(w)n(e)p Black 1922 5173 a(50)p Black eop end end %%Page: 51 51 TeXDict begin HPSdict begin 51 50 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.51) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(obtain)749 708 y(E)806 720 y Ft(Z)859 708 y FD([)p Fv(m)955 720 y Ft(d)993 708 y FD(\()p Fv(Z)q(;)14 b(\015)5 b FD(\)])23 b Fs(\024)1334 597 y FC(1)p Ft(=\015)t Fq(\000)p FC(1)1369 629 y Fu(X)1376 806 y Ft(i)p FC(=0)1538 708 y FD(E)1595 720 y Ft(Z)1648 708 y FD([)p Fv(m)1744 720 y Ft(d)1783 708 y FD(\()p Fv(Z)1872 720 y Ft(i)1900 708 y Fv(;)14 b(\015)5 b FD(\)])23 b Fs(\024)g FD(1)p Fv(=\015)f Fs(\001)c Fv(d)2384 566 y Fu( )2450 708 y FD(1)g Fs(\000)2603 583 y(p)p 2672 583 92 4 v 69 x FD(2)p Fv(\031)p 2603 689 161 4 v 2646 765 a(e)2685 741 y FC(2)2792 708 y Fs(\001)2844 591 y(p)p 2913 591 191 4 v 61 x FD(1)g Fs(\000)g Fv(\015)p 2844 689 260 4 v 2918 706 a Fs(p)p 2987 706 44 4 v 70 x Fv(d)3114 566 y Fu(!)2063 993 y Fs(\024)23 b Fv(n)2201 851 y Fu( )2266 993 y FD(1)18 b Fs(\000)2419 868 y(p)p 2488 868 92 4 v 69 x FD(2)p Fv(\031)p 2419 974 161 4 v 2462 1050 a(e)2501 1026 y FC(2)2608 993 y Fs(\001)2695 937 y FD(1)p 2660 974 113 4 v 2660 990 a Fs(p)p 2729 990 44 4 v 71 x Fv(d)2782 851 y Fu(!)2063 1212 y FD(=)23 b Fv(n)p FD(\(1)18 b Fs(\000)g Fv(\002)r FD(\()2473 1137 y Fu(p)p 2557 1137 127 4 v 2557 1212 a FD(1)p Fv(=d)o FD(\)\))p Fv(;)523 1400 y FD(\014nishing)26 b(the)i(pro)r(of)f(of)g(the)h(theorem.)1577 b Fs(u)-55 b(t)648 1586 y FD(The)26 b(follo)n(wing)d(lemma)g(deals)i (with)h(the)h(follo)n(wing)c(exp)r(erimen)n(t:)h(T)-7 b(oss)26 b Fv(n)g FD(coins)f(with)523 1685 y(probabilit)n(y)31 b Fv(p)k FD(of)g(heads,)f(return)g(the)i(n)n(um)n(b)r(er)d(of)i(heads)g (if)f(the)h(n)n(um)n(b)r(er)f(of)h(heads)f Fs(\024)523 1785 y Fv(\026)i FD(=)g Fv(np)f FD(\(the)h(exp)r(ectation\),)f (otherwise)e(return)i Fv(\026)p FD(.)h(What)g(is)e(the)i(exp)r(ected)g (v)-5 b(alue)34 b(of)523 1885 y(this)27 b(exp)r(erimen)n(t)e(relativ)n (e)f(to)j Fv(\026)p FD(?)g(The)h(probabilit)n(y)23 b(that)k(the)h (outcome)e(exceeds)g Fv(\026)18 b FD(+)g Fv(t\033)523 1984 y FD(decreases)33 b(exp)r(onen)n(tially)f(with)i Fv(t)p FD(,)i(where)e Fv(\033)39 b FD(is)34 b(the)h(standard)f (deviation)f(\()3026 1933 y Fs(p)p 3095 1933 51 4 v 51 x Fv(\026)i FD(in)g(our)523 2084 y(case\).)g(This)f(suggests)g(that)i (the)f(exp)r(ected)h(di\013erence)e(is)g Fv(O)r FD(\(1\))i(standard)f (deviations,)523 2184 y(whic)n(h)27 b(is)f(the)i(claim)c(of)k(the)g (next)g(Lemma:)523 2184 y SDict begin H.S end 523 2184 a 523 2184 a SDict begin 12 H.A end 523 2184 a 523 2184 a SDict begin [/View [/XYZ H.V]/Dest (lemma.28) cvn /DEST pdfmark end 523 2184 a Black 188 x Fr(Lemma)i(28.)p Black 40 w Fp(F)-6 b(or)30 b(any)g Fv(n)23 b Fs(2)g FD(N)p Fp(,)31 b FD(0)22 b Fv(<)h(p)g Fs(\024)g FD(1)p Fp(,)29 b(and)i Fv(\026)23 b FD(=)f Fv(np)p Fp(,)30 b(we)g(have)h(that)722 2631 y Fv(\026)772 2514 y Fu(\022)833 2631 y FD(1)18 b Fs(\000)1013 2574 y Fv(e)p 986 2612 92 4 v 986 2688 a FD(2)p Fv(\031)1098 2514 y Fs(p)p 1167 2514 185 4 v 60 x FD(1)g Fs(\000)g Fv(p)p 1098 2612 254 4 v 1165 2636 a Fs(p)p 1235 2636 51 4 v 1235 2688 a Fv(\026)1362 2514 y Fu(\023)1446 2631 y Fs(\024)23 b FD(E[)o(min)13 b(\()p Fv(B)t(in)p FD(\()p Fv(n;)h(p)p FD(\))p Fv(;)g(\026)p FD(\)])23 b Fs(\024)g Fv(\026)2440 2489 y Fu( )2505 2631 y FD(1)18 b Fs(\000)2658 2506 y(p)p 2727 2506 92 4 v 68 x FD(2)p Fv(\031)p 2658 2612 161 4 v 2701 2688 a(e)2740 2664 y FC(2)2839 2514 y Fs(p)p 2908 2514 185 4 v 60 x FD(1)g Fs(\000)g Fv(p)p 2839 2612 254 4 v 2906 2636 a Fs(p)p 2976 2636 51 4 v 2976 2688 a Fv(\026)3103 2489 y Fu(!)3183 2631 y Fv(:)p Black 523 2892 a Fp(Pr)l(o)l(of.)p Black 676 3080 a SDict begin H.S end 676 3080 a 676 3080 a SDict begin 12 H.A end 676 3080 a 676 3080 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.102) cvn /DEST pdfmark end 676 3080 a FD(E[)o(min)13 b(\()p Fv(B)t(in)p FD(\()p Fv(n;)h(p)p FD(\))p Fv(;)g(\026)p FD(\)])699 3271 y(=)827 3168 y Ft(n)787 3193 y Fu(X)787 3371 y Ft(k)q FC(=0)921 3271 y FD(min)f(\()p Fv(\026;)h(k)s FD(\))k Fs(\001)1331 3154 y Fu(\022)1392 3215 y Fv(n)1394 3328 y(k)1442 3154 y Fu(\023)1503 3271 y Fv(p)1545 3237 y Ft(k)1586 3271 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1835 3237 y Ft(n)p Fq(\000)p Ft(k)699 3554 y FD(=)827 3446 y Ft(\026)787 3475 y Fu(X)787 3654 y Ft(k)q FC(=0)921 3554 y Fv(k)967 3437 y Fu(\022)1028 3498 y Fv(n)1030 3611 y(k)1078 3437 y Fu(\023)1139 3554 y Fv(p)1181 3519 y Ft(k)1222 3554 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1471 3519 y Ft(n)p Fq(\000)p Ft(k)1651 3554 y FD(+)46 b Fv(\026)18 b Fs(\001)h FD(Prob)o([)13 b Fv(B)t(in)p FD(\()p Fv(n;)h(p)p FD(\))23 b Fs(\025)g Fv(\026)18 b FD(+)g(1)c(])699 3836 y(=)23 b Fv(\026)891 3728 y Ft(\026)851 3757 y Fu(X)851 3936 y Ft(k)q FC(=1)997 3780 y Fv(k)p 995 3817 51 4 v 995 3893 a(\026)1074 3836 y Fs(\001)1125 3780 y Fv(n)p 1125 3817 50 4 v 1127 3893 a(k)1185 3719 y Fu(\022)1246 3780 y Fv(n)18 b Fs(\000)g FD(1)1248 3893 y Fv(k)j Fs(\000)d FD(1)1439 3719 y Fu(\023)1500 3836 y Fv(p)1542 3802 y Ft(k)q Fq(\000)p FC(1)1668 3836 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1917 3802 y Ft(n)p Fq(\000)p Ft(k)2051 3836 y Fv(p)46 b FD(+)f Fv(\026)19 b Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))23 b Fs(\025)f Fv(\026)d FD(+)f(1)c(])699 4131 y(=)23 b Fv(\026)851 4023 y Ft(\026)p Fq(\000)p FC(1)853 4052 y Fu(X)853 4231 y Ft(k)q FC(=1)1000 4075 y Fv(np)p 1000 4112 92 4 v 1021 4188 a(\026)1120 4131 y Fs(\001)1161 4014 y Fu(\022)1223 4075 y Fv(n)18 b Fs(\000)g FD(1)1296 4188 y Fv(k)1415 4014 y Fu(\023)1476 4131 y Fv(p)1518 4097 y Ft(k)1559 4131 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1808 4097 y Ft(n)p Fq(\000)p FC(1)p Fq(\000)p Ft(k)2073 4131 y FD(+)46 b Fv(\026)18 b Fs(\001)h FD(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))22 b Fs(\025)h Fv(\026)18 b FD(+)g(1)c(])699 4335 y(=)23 b Fv(\026)18 b Fs(\001)h FD(Prob)o([)13 b Fv(B)t(in)p FD(\()p Fv(n)19 b Fs(\000)f FD(1)p Fv(;)c(p)p FD(\))22 b Fs(\024)h Fv(\026)18 b Fs(\000)g FD(1)c(])46 b(+)f Fv(\026)19 b Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))23 b Fs(\025)f Fv(\026)d FD(+)f(1)c(])p Black 123 w(\(102\))p Black 648 4526 a(W)-7 b(e)23 b(describ)r(e)e(a)h (single)e(exp)r(erimen)n(t)h(with)h(t)n(w)n(o)g(disjoin)n(t)f(ev)n(en)n (ts)h Fv(A)p FD(,)h(and)f Fv(B)t FD(,)h(suc)n(h)g(that)523 4625 y(Prob)o([)14 b Fv(A)g FD(])23 b(=)f(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b Fs(\024)f Fv(\026)d Fs(\000)f FD(1)c(])o(,)19 b(and)f(Prob)o([)c Fv(B)j FD(])24 b(=)e(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))22 b Fs(\025)h Fv(\026)c FD(+)f(1)c(])o(.)523 4725 y(This)23 b(will)e(b)r(e)j(useful)f (as)g(then)h(the)g(probabilit)n(y)19 b(of)24 b(\()p Fv(A)g FD(or)f Fv(B)t FD(\))h(is)f(simply)e(the)j(sum)e(of)i(these)523 4825 y(probabilities.)g(The)29 b(exp)r(erimen)n(t)e(is)g(to)i(toss)f Fv(n)h FD(coins,)e(where)i(the)g(probabilit)n(y)c(of)k(heads)523 4924 y(is)e Fv(p)p FD(,)p Black 1922 5173 a(51)p Black eop end end %%Page: 52 52 TeXDict begin HPSdict begin 52 51 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.52) cvn /DEST pdfmark end 515 125 a Black Black Black 575 448 a Fr({)p Black 41 w FD(Ev)n(en)n(t)31 b Fv(A)g FD(o)r(ccurs)g(if)g(amongst)e(the)j(\014rst)f Fv(n)21 b Fs(\000)g FD(1)31 b(results)e(there)j(w)n(ere)e(no)h(more)f (than)664 548 y Fv(\026)19 b Fs(\000)f FD(1)27 b(heads,)g(Prob)o([)14 b Fv(A)g FD(])23 b(=)f(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b Fs(\024)f Fv(\026)d Fs(\000)f FD(1)c(])o(.)p Black 575 652 a Fr({)p Black 41 w FD(Ev)n(en)n(t)23 b Fv(B)k FD(o)r(ccurs)c(if)f(in)h(total,)f (o)n(v)n(er)g Fv(n)h FD(coin)f(tosses,)g(there)i(w)n(ere)e(at)h(least)f Fv(\026)10 b FD(+)g(1)22 b(heads,)664 752 y(Prob)o([)14 b Fv(B)k FD(])23 b(=)f(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))23 b Fs(\025)f Fv(\026)d FD(+)f(1)c(])o(.)p Black 575 856 a Fr({)p Black 41 w FD(It)32 b(is)f(easy)g(to)g(see)g (that)h(if)g Fv(A)g FD(holds)e(then)i Fv(B)k FD(cannot)31 b(o)r(ccur.)g(Lik)n(ewise,)e(if)i Fv(B)36 b FD(holds)664 955 y(then)28 b Fv(A)g FD(cannot)f(o)r(ccur.)648 1135 y(Th)n(us,)893 1326 y SDict begin H.S end 893 1326 a 893 1326 a SDict begin 12 H.A end 893 1326 a 893 1326 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.103) cvn /DEST pdfmark end 893 1326 a FD(Prob)o([)14 b Fv(A)19 b Fs(_)f Fv(B)g FD(])24 b(=)e(Prob)o([)14 b Fv(A)g FD(])k(+)g(Prob)o([)c Fv(B)k FD(])917 1451 y(=)k(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b Fs(\024)f Fv(\026)d Fs(\000)f FD(1)c(])k(+)g(Prob)n([)c Fv(B)t(in)p FD(\()p Fv(n;)g(p)p FD(\))23 b Fs(\025)g Fv(\026)18 b FD(+)g(1)c(])p Black 181 w(\(103\))p Black 523 1642 a(On)28 b(the)h(other)e(hand,)i(the)f(complimen)n(tary)c(ev)n (en)n(t)j(to)h Fv(A)20 b Fs(_)f Fv(B)t FD(,)p 2562 1575 222 4 v 28 w Fv(A)g Fs(_)g Fv(B)t FD(,)28 b(o)r(ccurs)g(if)f(during)523 1742 y(the)e(\014rst)g Fv(n)13 b Fs(\000)g FD(1)23 b(tosses)g(there)i (w)n(ere)f(exactly)f Fv(\026)i FD(heads)f(and)g(the)h(last)f(coin)f (toss)h(ga)n(v)n(e)f(tails.)523 1841 y(Therefore)758 2033 y SDict begin H.S end 758 2033 a 758 2033 a SDict begin 12 H.A end 758 2033 a 758 2033 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.104) cvn /DEST pdfmark end 758 2033 a FD(Prob)o([)14 b Fv(A)k Fs(_)h Fv(B)f FD(])23 b(=)g(1)18 b Fs(\000)g FD(Prob)1661 1965 y Fu(\002)p 1737 1966 222 4 v 1737 2033 a Fv(A)h Fs(_)f Fv(B)2000 1965 y Fu(\003)1253 2157 y FD(=)23 b(1)18 b Fs(\000)g FD(Prob)o([)c Fv(B)t(in)p FD(\(1)p Fv(;)g(p)p FD(\))22 b(=)h(0)14 b(])k Fs(\001)g FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))22 b(=)h Fv(\026)14 b FD(])1253 2282 y(=)23 b(1)18 b Fs(\000)g FD(\(1)g Fs(\000)g Fv(p)p FD(\))h Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b(=)f Fv(\026)14 b FD(])p Black 530 w(\(104\))p Black 523 2473 a(Substituting)27 b(\()1031 2473 y SDict begin H.S end 1031 2473 a Black FD(104)p Black 1155 2420 a SDict begin H.R end 1155 2420 a 1155 2473 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.104) cvn H.B /ANN pdfmark end 1155 2473 a FD(\))h(in)n(to)f(\()1416 2473 y SDict begin H.S end 1416 2473 a Black FD(103)p Black 1541 2420 a SDict begin H.R end 1541 2420 a 1541 2473 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.103) cvn H.B /ANN pdfmark end 1541 2473 a FD(\))g(and)h(the)g(result)e(in)n(to)h(\()2337 2473 y SDict begin H.S end 2337 2473 a Black FD(102)p Black 2461 2420 a SDict begin H.R end 2461 2420 a 2461 2473 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.102) cvn H.B /ANN pdfmark end 2461 2473 a FD(\))h(w)n(e)g(obtain)694 2665 y SDict begin H.S end 694 2665 a 694 2665 a SDict begin 12 H.A end 694 2665 a 694 2665 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.105) cvn /DEST pdfmark end 694 2665 a FD(E[min)12 b(\()q Fv(B)t(in)p FD(\()p Fv(n;)i(p)p FD(\))p Fv(;)g(\026)p FD(\))o(])24 b(=)e Fv(\026)1600 2597 y Fu(\000)1638 2665 y FD(1)c Fs(\000)g FD(\(1)g Fs(\000)g Fv(p)p FD(\))h Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b(=)f Fv(\026)14 b FD(])2983 2597 y Fu(\001)3021 2665 y Fv(:)172 b FD(\(105\))523 2856 y(T)-7 b(o)20 b(\014nish)f(the)h(pro)r(of)f(w)n (e)h(b)r(ound)g(\(1)s Fs(\000)s Fv(p)p FD(\))s Fs(\001)s FD(Prob)o([)14 b Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b(=)f Fv(\026)14 b FD(],)20 b(recall)d(that)k Fv(\026)i FD(=)f Fv(np)p FD(:)523 3048 y SDict begin H.S end 523 3048 a 523 3048 a SDict begin 12 H.A end 523 3048 a 523 3048 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.106) cvn /DEST pdfmark end 523 3048 a FD(\(1)c Fs(\000)g Fv(p)p FD(\))h Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))23 b(=)f Fv(\026)14 b FD(])546 3222 y(=)23 b(\(1)18 b Fs(\000)g Fv(p)p FD(\))883 3105 y Fu(\022)945 3166 y Fv(n)g Fs(\000)g FD(1)995 3279 y Fv(np)1137 3105 y Fu(\023)1198 3222 y Fv(p)1240 3188 y Ft(np)1320 3222 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1569 3188 y Ft(n)p Fq(\000)p FC(1)p Fq(\000)p Ft(np)546 3453 y FD(=)860 3396 y(\()p Fv(n)h Fs(\000)f FD(1\)!)p 644 3433 713 4 v 644 3509 a(\()p Fv(np)p FD(\)!)h Fs(\001)f FD(\()p Fv(n)h Fs(\000)f FD(1)g Fs(\000)g Fv(np)p FD(\)!)1367 3453 y Fv(p)1409 3418 y Ft(np)1488 3453 y FD(\(1)g Fs(\000)g Fv(p)p FD(\))1737 3418 y Ft(n)p Fq(\000)p Ft(np)546 3677 y FD(=)644 3621 y Fv(n)g Fs(\000)g Fv(np)p 644 3658 243 4 v 741 3734 a(n)915 3677 y Fs(\001)1215 3621 y Fv(n)p FD(!)p 967 3658 570 4 v 967 3734 a(\()p Fv(np)p FD(\)!)h Fs(\001)f FD(\()p Fv(n)h Fs(\000)f Fv(np)p FD(\)!)1565 3677 y Fs(\001)h Fv(p)1649 3643 y Ft(np)1728 3677 y FD(\(1)f Fs(\000)g Fv(p)p FD(\))1977 3643 y Ft(n)p Fq(\000)p Ft(np)546 3920 y Fs(\025)644 3864 y Fv(n)g Fs(\000)g Fv(np)p 644 3901 243 4 v 741 3977 a(n)915 3920 y Fs(\001)967 3795 y(p)p 1036 3795 92 4 v 69 x FD(2)p Fv(\031)j Fs(\001)e Fv(n)1238 3833 y Ft(n)1283 3804 y Fs(p)p 1352 3804 50 4 v 60 x Fv(n)p 967 3901 436 4 v 1142 3977 a(e)1181 3953 y Ft(n)800 4132 y Fs(\001)1065 4076 y Fv(e)1104 4046 y Ft(np)p 847 4113 555 4 v 847 4189 a Fv(e)f Fs(\001)h FD(\()p Fv(np)p FD(\))1102 4165 y Ft(np)1200 4189 y Fs(\001)1241 4137 y(p)p 1310 4137 92 4 v 52 x Fv(np)1444 4132 y Fs(\001)1879 4076 y Fv(e)1918 4046 y Ft(n)p Fq(\000)p Ft(np)p 1510 4113 951 4 v 1510 4189 a Fv(e)f Fs(\001)g FD(\()p Fv(n)h Fs(\000)f Fv(np)p FD(\))1916 4165 y Ft(n)p Fq(\000)p Ft(np)2106 4189 y Fs(\001)2148 4131 y(p)p 2217 4131 243 4 v 58 x Fv(n)g Fs(\000)g Fv(np)2488 4132 y Fs(\001)h Fv(p)2572 4098 y Ft(np)2651 4132 y FD(\(1)f Fs(\000)g Fv(p)p FD(\))2900 4098 y Ft(n)p Fq(\000)p Ft(np)p Black 3318 4132 a FD(\(106\))p Black 546 4382 a(=)644 4257 y Fs(p)p 713 4257 92 4 v 69 x FD(2)p Fv(\031)p 644 4363 161 4 v 687 4439 a(e)726 4415 y FC(2)833 4382 y Fs(\001)885 4326 y Fv(n)g Fs(\000)g Fv(np)p 885 4363 243 4 v 981 4439 a(n)1156 4382 y Fs(\001)1415 4266 y(p)p 1484 4266 50 4 v 60 x Fv(n)p 1208 4363 533 4 v 1208 4387 a Fs(p)p 1277 4387 92 4 v 52 x Fv(np)g Fs(\001)1428 4381 y(p)p 1498 4381 243 4 v 1498 4439 a Fv(n)g Fs(\000)g Fv(np)1769 4382 y Fs(\001)2310 4326 y Fv(n)2360 4295 y Ft(n)p 1820 4363 1075 4 v 1820 4439 a Fv(n)1870 4415 y Ft(np)1968 4439 y Fs(\001)h Fv(p)2052 4415 y Ft(np)2149 4439 y Fs(\001)g Fv(n)2241 4415 y Ft(n)p Fq(\000)p Ft(np)2431 4439 y Fs(\001)g FD(\(1)f Fs(\000)g Fv(p)p FD(\))2722 4415 y Ft(n)p Fq(\000)p Ft(np)2923 4382 y Fs(\001)h Fv(p)3007 4347 y Ft(np)3086 4382 y FD(\(1)f Fs(\000)g Fv(p)p FD(\))3335 4347 y Ft(n)p Fq(\000)p Ft(np)546 4631 y FD(=)644 4506 y Fs(p)p 713 4506 92 4 v 69 x FD(2)p Fv(\031)p 644 4612 161 4 v 687 4688 a(e)726 4664 y FC(2)833 4631 y Fs(\001)885 4575 y Fv(n)g Fs(\000)g Fv(np)p 885 4612 243 4 v 981 4688 a(n)1156 4631 y Fs(\001)1415 4515 y(p)p 1484 4515 50 4 v 60 x Fv(n)p 1208 4612 533 4 v 1208 4637 a Fs(p)p 1277 4637 92 4 v 51 x Fv(np)g Fs(\001)1428 4630 y(p)p 1498 4630 243 4 v 1498 4688 a Fv(n)g Fs(\000)g Fv(np)546 4881 y FD(=)644 4756 y Fs(p)p 713 4756 92 4 v 69 x FD(2)p Fv(\031)p 644 4862 161 4 v 687 4938 a(e)726 4914 y FC(2)833 4881 y Fs(\001)885 4764 y(p)p 954 4764 185 4 v 61 x FD(1)g Fs(\000)g Fv(p)p 885 4862 254 4 v 952 4886 a Fs(p)p 1021 4886 51 4 v 52 x Fv(\026)p Black 3318 4881 a FD(\(107\))p Black Black 1922 5173 a(52)p Black eop end end %%Page: 53 53 TeXDict begin HPSdict begin 53 52 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 515 125 a SDict begin H.S end 515 125 a 515 125 a SDict begin H.R end 515 125 a 515 125 a SDict begin [/View [/XYZ H.V]/Dest (page.53) cvn /DEST pdfmark end 515 125 a Black Black 523 448 a FD(where)19 b(Inequalit)n(y)f(\()1172 448 y SDict begin H.S end 1172 448 a Black FD(106)p Black 1297 395 a SDict begin H.R end 1297 395 a 1297 448 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.106) cvn H.B /ANN pdfmark end 1297 448 a FD(\))i(follo)n(ws)d(from)h(Stirling's)f (lo)n(w)n(er)g(b)r(ound)k(on)e Fv(n)p FD(!)h(in)f(the)i(n)n(umerator) 523 548 y(and)27 b(Stirling's)e(upp)r(er)i(b)r(ound)h(on)g(\()p Fv(np)p FD(\)!)g(and)f(\()p Fv(n)19 b Fs(\000)f Fv(np)p FD(\)!)27 b(in)g(the)h(den)n(umerator.)648 648 y(A)21 b(similar)16 b(pro)r(of,)k(using)f(Stirling's)f(upp)r(er)j(b)r(ound)g (on)g Fv(n)p FD(!)f(and)h(Stirling's)c(lo)n(w)n(er)i(b)r(ound)523 747 y(on)27 b(\()p Fv(np)p FD(\)!)h(and)g(\()p Fv(n)18 b Fs(\000)g Fv(np)p FD(\)!)28 b(yields)1115 983 y SDict begin H.S end 1115 983 a 1115 983 a SDict begin 12 H.A end 1115 983 a 1115 983 a SDict begin [/View [/XYZ H.V]/Dest (equation.A.4.108) cvn /DEST pdfmark end 1115 983 a FD(\(1)18 b Fs(\000)g Fv(p)p FD(\))h Fs(\001)f FD(Prob)o([)c Fv(B)t(in)p FD(\()p Fv(n)k Fs(\000)g FD(1)p Fv(;)c(p)p FD(\))22 b(=)h Fv(\026)14 b FD(])23 b Fs(\024)2464 927 y Fv(e)p 2438 964 92 4 v 2438 1040 a FD(2)p Fv(\031)2549 866 y Fs(p)p 2618 866 185 4 v 61 x FD(1)18 b Fs(\000)g Fv(p)p 2549 964 254 4 v 2617 988 a Fs(p)p 2686 988 51 4 v 52 x Fv(\026)3216 983 y FD(\(108\))523 1221 y(Assigning)24 b(b)r(ound)j(\()1186 1221 y SDict begin H.S end 1186 1221 a Black FD(108)p Black 1311 1168 a SDict begin H.R end 1311 1168 a 1311 1221 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.108) cvn H.B /ANN pdfmark end 1311 1221 a FD(\))g(and)f(\()1562 1221 y SDict begin H.S end 1562 1221 a Black FD(107)p Black 1687 1168 a SDict begin H.R end 1687 1168 a 1687 1221 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.106) cvn H.B /ANN pdfmark end 1687 1221 a FD(\))h(in)n(to)e(equation)g(\()2283 1221 y SDict begin H.S end 2283 1221 a Black FD(105)p Black 2408 1168 a SDict begin H.R end 2408 1168 a 2408 1221 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.A.4.105) cvn H.B /ANN pdfmark end 2408 1221 a FD(\))i(giv)n(es)d(the)j(statemen)n(t)f (of)h(the)523 1321 y(lemma.)2561 b Fs(u)-55 b(t)p Black 1922 5173 a FD(53)p Black eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF