%!PS-Adobe-2.0 %%Creator: dvips(k) 5.996 Copyright 2016 Radical Eye Software %%Title: arXiv:1901.06639v2 [math.FA] 9 Apr 2020 %%CreationDate: Mon Apr 13 13:24:54 2020 %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z 20200406-random_info.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2020.04.13:1324 %%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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b(26])g(for)g(a)f(presen)m(tation)i(of)e(these)j(results.)54 b(Indeed,)38 b(based)f(on)f(the)h(pro)s(of)e(tec)m(hnique)j(of)-9 5622 y(the)29 b(presen)m(t)h(pap)s(er,)g(it)e(w)m(as)h(recen)m(tly)h (pro)m(v)m(en)g(in)e([)1880 5622 y SDict begin H.S end 1880 5622 a 0 TeXcolorgray Fr(16)p 0 TeXcolorgray 1978 5558 a SDict begin H.R end 1978 5558 a 1978 5622 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.KU19) cvn H.B /ANN pdfmark end 1978 5622 a Fr(])h(that)f(the)h(p)s(olynomial)c(order)k(of)f(con)m(v) m(ergence)-9 5767 y(for)k(function)g(v)-5 b(alues)32 b(and)h(general)f(linear)e(information)g(is)i(the)h(same)f(for)g(all)f Fm(\033)g Fp(2)d Fm(`)3153 5782 y Fl(2)3193 5767 y Fr(.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 TeXcolorgray 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 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1375 a -9 1375 a SDict begin 17.37396 H.A end -9 1375 a -9 1375 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end -9 1375 a 1284 1500 a Fr(2.)49 b Fq(Pr)m(oblem)37 b(and)f(resul)-8 b(ts)-9 1703 y Fr(W)g(e)32 b(consider)h(the)g (ellipsoid)1192 1967 y Fp(E)1245 1982 y Fn(\033)1319 1967 y Fk(=)1423 1797 y Fj(\()1503 1967 y Fm(x)28 b Fp(2)g Fm(`)1721 1982 y Fl(2)1771 1967 y Fk(:)1848 1873 y Fj(X)1857 2085 y Fn(j)t Fi(2)p Fg(N)2009 1827 y Fj(\022)2092 1900 y Fm(x)2147 1915 y Fn(j)p 2092 1944 92 4 v 2092 2036 a Fm(\033)2147 2051 y Fn(j)2194 1827 y Fj(\023)2267 1849 y Fl(2)2335 1967 y Fp(\024)g Fk(1)2489 1797 y Fj(\))-9 2260 y Fr(with)22 b(semi-axes)g(of)g(lengths)g Fm(\033)1112 2275 y Fl(1)1179 2260 y Fp(\025)29 b Fm(\033)1340 2275 y Fl(2)1407 2260 y Fp(\025)f(\001)17 b(\001)g(\001)26 b(\025)i Fk(0)p Fr(.)1837 2260 y SDict begin H.S end 1837 2260 a -39 x Fe(1)1873 2260 y SDict begin 17.37396 H.L end 1873 2260 a 1873 2260 a SDict begin [/Subtype /Link/Dest (Hfootnote.1) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [1 0 0] H.B /ANN pdfmark end 1873 2260 a 40 w Fr(This)22 b(is)g(the)h(unit)f(ball)e(of)i(a)g(Hilb)s(ert)f(space,)26 b(whic)m(h)-9 2404 y(w)m(e)32 b(denote)h(b)m(y)f Fm(H)662 2419 y Fn(\033)709 2404 y Fr(.)43 b(W)-8 b(e)32 b(study)h(the)f (problem)e(of)h(reco)m(v)m(ering)i(an)e(unkno)m(wn)j(v)m(ector)e Fm(x)c Fp(2)g(E)3404 2419 y Fn(\033)3482 2404 y Fr(from)j Fm(n)-9 2548 y Fr(pieces)37 b(of)f(information,)e(where)k(w)m(e)g(w)m (an)m(t)f(to)f(guaran)m(tee)h(a)g(small)d(error)i(in)g Fm(`)2948 2563 y Fl(2)2987 2548 y Fr(.)56 b(The)37 b(information)-9 2693 y(ab)s(out)43 b Fm(x)48 b Fp(2)g(E)551 2708 y Fn(\033)642 2693 y Fr(is)43 b(giv)m(en)h(b)m(y)i(the)e(outcome)g Fm(L)1812 2708 y Fl(1)1852 2693 y Fk(\()p Fm(x)p Fk(\))p Fm(;)17 b(:)g(:)g(:)f(;)h(L)2268 2708 y Fn(n)2315 2693 y Fk(\()p Fm(x)p Fk(\))44 b Fr(of)g Fm(n)g Fr(linear)f(functionals.) 3490 2693 y SDict begin H.S end 3490 2693 a -39 x Fe(2)3524 2693 y SDict begin 17.37396 H.L end 3524 2693 a 3524 2693 a SDict begin [/Subtype /Link/Dest (Hfootnote.2) cvn/H /I/Border [0 0 0]BorderArrayPatch/Color [1 0 0] H.B /ANN pdfmark end 3524 2693 a 78 w Fr(The)-9 2837 y(mapping)29 b Fm(N)468 2852 y Fn(n)543 2837 y Fk(=)e(\()p Fm(L)750 2852 y Fl(1)790 2837 y Fm(;)17 b(:)g(:)g(:)f(;)h(L)1075 2852 y Fn(n)1122 2837 y Fk(\))31 b Fr(is)g(called)f(the)h(information)d (mapping.)41 b(A)32 b(reco)m(v)m(ery)h(algorithm)28 b(is)i(a)-9 2981 y(mapping)k Fm(A)468 2996 y Fn(n)548 2981 y Fk(:)g Fm(H)690 2996 y Fn(\033)770 2981 y Fp(!)f Fm(`)944 2996 y Fl(2)1019 2981 y Fr(of)i(the)i(form)d Fm(A)1611 2996 y Fn(n)1692 2981 y Fk(=)f Fm(')24 b Fp(\016)g Fm(N)2041 2996 y Fn(n)2089 2981 y Fr(,)36 b(where)h Fm(')f Fr(maps)g Fm(N)2874 2996 y Fn(n)2921 2981 y Fk(\()p Fm(x)p Fk(\))g Fr(bac)m(k)h(to)e Fm(`)3482 2996 y Fl(2)3522 2981 y Fr(.)53 b(The)-9 3125 y(w)m(orst)33 b(case)g(error)g(of)f(the)h(algorithm)c(is) j(giv)m(en)h(b)m(y)1297 3332 y Fm(e)p Fk(\()p Fm(A)1453 3347 y Fn(n)1501 3332 y Fk(\))27 b(=)37 b(sup)1670 3414 y Fn(x)p Fi(2E)1794 3422 y Ff(\033)1852 3332 y Fp(k)p Fm(A)1975 3347 y Fn(n)2022 3332 y Fk(\()p Fm(x)p Fk(\))22 b Fp(\000)h Fm(x)p Fp(k)2380 3361 y Fl(2)2436 3332 y Fm(:)-9 3571 y Fr(The)29 b(qualit)m(y)e(of)h(the)h(information)c (mapping)i(is)g(measured)i(b)m(y)g(its)f(radius,)h(whic)m(h)g(is)e(the) i(w)m(orst)g(case)-9 3716 y(error)j(of)g(the)h(b)s(est)g(reco)m(v)m (ery)i(algorithm)29 b(based)34 b(on)e(the)h(information)c Fm(N)2727 3731 y Fn(n)2774 3716 y Fr(,)k(i.e.,)1103 3922 y Fk(rad\()p Fm(N)1360 3937 y Fn(n)1407 3922 y Fm(;)17 b Fp(E)1504 3937 y Fn(\033)1550 3922 y Fk(\))28 b(:=)200 b(inf)1747 3989 y Fn(')8 b Fl(:)23 b(im\()p Fn(N)2005 3997 y Ff(n)2048 3989 y Fl(\))p Fi(!)p Fn(`)2175 3998 y Fd(2)2226 3922 y Fm(e)p Fk(\()p Fm(')f Fp(\016)g Fm(N)2545 3937 y Fn(n)2592 3922 y Fk(\))p Fm(:)-9 4161 y Fr(Note)34 b(that)g(this)f(is)h(a)g(linear)e(problem)h(o)m(v)m(er)j(Hilb)s(ert)c (spaces)k(as)e(describ)s(ed)h(in)e([)3013 4161 y SDict begin H.S end 3013 4161 a 0 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b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(11)p 0 TeXcolorgray -9 121 a Fr(T)-8 b(urning)35 b(to)g(the)h(in\034nite)e(dimensional)f(case,)k(w)m(e)g(note)f(that)f (w)m(e)h(ha)m(v)m(e)h Fm(s)2769 136 y Fl(1)2809 121 y Fk(\()p Fm(A)p Fk(\))32 b Fm(>)h(C)42 b Fr(if)35 b(and)g(only)g(if)-9 265 y(there)e(is)f(some)g Fm(m)c Fp(2)g Fo(N)48 b Fr(suc)m(h)34 b(that)e Fm(s)1371 280 y Fl(1)1410 265 y Fk(\()p Fm(A)1521 280 y Fn(m)1588 265 y Fk(\))c Fm(>)f(C)7 b Fr(.)44 b(This)32 b(yields)352 472 y Fo(P)p Fk([)p Fm(s)484 487 y Fl(1)526 472 y Fk(\()p Fm(A)p Fk(\))27 b Fm(>)h(C)7 b Fk(])28 b(=)f Fo(P)17 b Fk([)r Fp(9)p Fm(m)28 b Fp(2)g Fo(N)20 b Fk(:)39 b Fm(s)1597 487 y Fl(1)1637 472 y Fk(\()p Fm(A)1748 487 y Fn(m)1815 472 y Fk(\))27 b Fm(>)h(C)7 b Fk(])27 b(=)62 b(lim)2219 532 y Fn(m)p Fi(!1)2439 472 y Fo(P)p Fk([)p Fm(s)2571 487 y Fl(1)2612 472 y Fk(\()p Fm(A)2723 487 y Fn(m)2790 472 y Fk(\))28 b Fm(>)f(C)7 b Fk(])28 b Fp(\024)g Fm(e)3241 431 y Fi(\000)p Fn(c)3327 407 y Fd(2)3361 431 y Fn(n)-9 687 y Fr(since)k Fm(s)275 702 y Fl(1)315 687 y Fk(\()p Fm(A)426 702 y Fn(m)493 687 y Fk(\))g Fr(is)g(increasing)g(in)f Fm(m)i Fr(and)g Fm(C)i Fp(\025)28 b Fm(C)1818 702 y Fn(m)1884 687 y Fr(.)1782 b Fc(\003)115 881 y Fr(T)-8 b(ogether)33 b(with)f(Prop)s(osition)1271 881 y SDict begin H.S end 1271 881 a 0 TeXcolorgray Fr(10)p 0 TeXcolorgray 1369 817 a SDict begin H.R end 1369 817 a 1369 881 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.10) cvn H.B /ANN pdfmark end 1369 881 a 32 w Fr(this)g(yields)g (that)h(the)g(estimate)-9 1194 y(\(2\))974 1194 y SDict begin H.S end 974 1194 a 974 1194 a SDict begin 17.37396 H.A end 974 1194 a 974 1194 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 974 1194 a Fm(e)p Fk(\()p Fm(A)1130 1209 y Fn(n)1177 1194 y Fk(\))28 b Fp(\024)g Fm(\033)1403 1209 y Fn(k)r Fl(+1)1558 1194 y Fk(+)1676 1048 y Fl(3)p 1676 1065 36 4 v 1676 1122 a(2)1721 980 y Fj(q)p 1821 980 351 4 v 33 x(P)1926 1117 y Fn(j)t(>k)2073 1087 y Fm(\033)2132 1053 y Fl(2)2128 1113 y Fn(j)2193 1087 y Fk(+)22 b(11)p Fm(c)17 b(\033)2503 1102 y Fn(k)r Fl(+1)2636 1016 y Fp(p)p 2719 1016 59 4 v 71 x Fm(n)p 1666 1171 1111 4 v 2040 1262 a(s)2086 1277 y Fn(k)2145 1262 y Fk(\()p Fm(G)2260 1277 y Fn(n;k)2365 1262 y Fk(\))-9 1453 y Fr(holds)37 b(with)h(probabilit)m(y)f(at)h (least)f Fk(1)26 b Fp(\000)h Fm(e)1566 1417 y Fi(\000)p Fn(c)1652 1393 y Fd(2)1686 1417 y Fn(n)1771 1453 y Fr(for)38 b(all)e Fm(k)41 b Fp(\024)d Fm(n)g Fr(and)g Fm(c)g Fp(\025)g Fk(1)p Fr(.)60 b(It)39 b(remains)e(to)h(b)s(ound)-9 1597 y(the)h Fm(k)s Fr(-th)g(singular)e(v)-5 b(alue)39 b(of)f(the)i (Gaussian)e(matrix)g Fm(G)2135 1612 y Fn(n;k)2279 1597 y Fr(from)g(b)s(elo)m(w.)62 b(It)40 b(is)e(kno)m(wn)j(from)c([)3645 1597 y SDict begin H.S end 3645 1597 a 0 TeXcolorgray Fr(35)p 0 TeXcolorgray 3743 1533 a SDict begin H.R end 3743 1533 a 3743 1597 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.RV09) cvn H.B /ANN pdfmark end 3743 1597 a Fr(,)-9 1741 y(Theorem)46 b(1.1])f(that)h(this)f(n)m(um)m(b)s(er)h(t)m (ypically)f(is)g(of)h(order)2324 1669 y Fp(p)p 2407 1669 59 4 v 72 x Fm(n)32 b Fp(\000)2606 1660 y(p)p 2689 1660 225 4 v 81 x Fm(k)25 b Fp(\000)d Fk(1)46 b Fr(for)f(all)f Fm(n)51 b Fp(2)f Fo(N)61 b Fr(and)-9 1885 y Fm(k)30 b Fp(\024)e Fm(n)p Fr(.)43 b(T)-8 b(o)31 b(exploit)e(our)h(upp)s(er)g(b)s (ound)h(to)f(full)e(extend,)k(the)f(n)m(um)m(b)s(er)f Fm(k)h Fp(\024)d Fm(n)j Fr(ma)m(y)e(b)s(e)i(c)m(hosen)h(suc)m(h)-9 2030 y(that)40 b(the)g(righ)m(t-hand)g(side)g(of)47 b(\()1240 2030 y SDict begin H.S end 1240 2030 a 0 TeXcolorgray Fr(2)p 0 TeXcolorgray 1289 1965 a SDict begin H.R end 1289 1965 a 1289 2030 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 1289 2030 a Fr(\))40 b(b)s(ecomes)h(minimal.)62 b(W)-8 b(e)41 b(realize)e(that)h(the)h(term) f Fk(1)p Fm(=s)3470 2045 y Fn(k)3512 2030 y Fk(\()p Fm(G)3627 2045 y Fn(n;k)3732 2030 y Fk(\))-9 2174 y Fr(increases)46 b(with)g Fm(k)s Fr(,)k(whereas)d(all)d(remaining)f(terms)j(decrease)i (with)e Fm(k)s Fr(.)84 b(Ho)m(w)m(ev)m(er,)51 b(the)c(in)m(v)m(erse)-9 2318 y(singular)28 b(n)m(um)m(b)s(er)i(ac)m(hiev)m(es)h(its)f(minimal) 25 b(order)30 b Fm(n)1903 2282 y Fi(\000)p Fl(1)p Fn(=)p Fl(2)2098 2318 y Fr(already)f(for)h Fm(k)g Fk(=)e Fm(cn)i Fr(with)f(some)h Fm(c)e Fp(2)g Fk(\(0)p Fm(;)17 b Fk(1\))p Fr(.)-9 2462 y(If)32 b Fm(\033)k Fr(do)s(es)c(not)g(deca)m(y)h (extremely)g(fast,)f(this)f(do)s(es)i(not)f(lead)f(to)h(a)g(loss)g (regarding)f(the)h(other)g(terms)-9 2607 y(of)43 b(\()151 2607 y SDict begin H.S end 151 2607 a 0 TeXcolorgray Fr(2)p 0 TeXcolorgray 199 2542 a SDict begin H.R end 199 2542 a 199 2607 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 199 2607 a Fr(\).)56 b(F)-8 b(or)35 b(instance,)j(w)m(e)g(ma)m(y)e(c)m(ho)s(ose)h Fm(k)h Fk(=)c Fp(b)p Fm(n=)p Fk(2)p Fp(c)j Fr(and)g(use)g(the)g(follo)m (wing)d(sp)s(ecial)i(case)h(of)f([)3694 2607 y SDict begin H.S end 3694 2607 a 0 TeXcolorgray Fr(6)p 0 TeXcolorgray 3743 2542 a SDict begin H.R end 3743 2542 a 3743 2607 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.DS01) cvn H.B /ANN pdfmark end 3743 2607 a Fr(,)-9 2751 y(Theorem)c(I)s(I.13].)-9 2951 y SDict begin H.S end -9 2951 a -9 2951 a SDict begin 17.37396 H.A end -9 2951 a -9 2951 a SDict begin [/View [/XYZ H.V]/Dest (thm.12) cvn /DEST pdfmark end -9 2951 a 0 TeXcolorgray FE(Lemma)39 b(12.)p 0 TeXcolorgray 41 w Fh(L)-5 b(et)35 b Fm(n)28 b Fp(2)g Fo(N)50 b Fh(and)34 b Fm(k)d Fk(=)d Fp(b)p Fm(n=)p Fk(2)p Fp(c)p Fh(.)45 b(Then)1211 3161 y Fo(P)1270 3051 y Fj(h)1319 3161 y Fm(s)1365 3176 y Fn(k)1425 3161 y Fk(\()o Fm(G)1539 3176 y Fn(n;k)1644 3161 y Fk(\))28 b Fp(\024)1815 3085 y(p)p 1898 3085 59 4 v 76 x Fm(n)q(=)p Fk(7)2055 3051 y Fj(i)2129 3161 y Fp(\024)g Fm(e)2279 3120 y Fi(\000)p Fn(n=)p Fl(100)2522 3161 y Fm(:)p 0 TeXcolorgray -9 3365 a Fh(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(It)32 b(is)g(sho)m(wn)i(in)e([)929 3365 y SDict begin H.S end 929 3365 a 0 TeXcolorgray Fr(6)p 0 TeXcolorgray 978 3301 a SDict begin H.R end 978 3301 a 978 3365 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.DS01) cvn H.B /ANN pdfmark end 978 3365 a Fr(,)g(Theorem)h(I)s(I.13]) f(that,)h(for)f(all)e Fm(k)h Fp(\024)d Fm(n)33 b Fr(and)g Fm(t)28 b(>)f Fk(0)p Fr(,)32 b(w)m(e)i(ha)m(v)m(e)895 3575 y Fo(P)971 3465 y Fj(h)1020 3575 y Fm(s)1066 3590 y Fn(k)1125 3575 y Fk(\()p Fm(G)1240 3590 y Fn(n;k)1345 3575 y Fk(\))28 b Fp(\024)1516 3499 y(p)p 1599 3499 59 4 v 76 x Fm(n)1674 3465 y Fj(\020)1733 3575 y Fk(1)22 b Fp(\000)1904 3485 y Fj(p)p 2003 3485 162 4 v 2003 3575 a Fm(k)s(=n)g Fp(\000)h Fm(t)2321 3465 y Fj(\021)q(i)2455 3575 y Fp(\024)29 b Fm(e)2606 3534 y Fi(\000)p Fn(nt)2729 3511 y Fd(2)2764 3534 y Fn(=)p Fl(2)2838 3575 y Fm(:)-9 3801 y Fr(The)k(statemen)m(t)g(follo)m(ws)e(b)m(y)j(putting)d Fm(k)g Fk(=)d Fp(b)p Fm(n=)p Fk(2)p Fp(c)33 b Fr(and)f Fm(t)2133 3765 y Fi(\000)p Fl(1)2255 3801 y Fk(=)2359 3719 y Fp(p)p 2442 3719 98 4 v 82 x Fk(50)p Fr(.)1126 b Fc(\003)115 3996 y Fr(If)44 b Fm(\033)k Fr(deca)m(ys)e(v)m(ery)g (fast,)h Fm(k)k Fk(=)c Fp(b)p Fm(n=)p Fk(2)p Fp(c)e Fr(migh)m(t)d(not)i (b)s(e)h(the)f(b)s(est)h(c)m(hoice.)79 b(The)45 b(term)e Fm(\033)3511 4011 y Fn(k)r Fl(+1)3689 3996 y Fr(in)-9 4140 y(estimate)35 b(\()423 4140 y SDict begin H.S end 423 4140 a 0 TeXcolorgray Fr(2)p 0 TeXcolorgray 472 4076 a SDict begin H.R end 472 4076 a 472 4140 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.2) cvn H.B /ANN pdfmark end 472 4140 a Fr(\))i(ma)m(y)f(b)s(e)h(m)m(uc)m(h)g(smaller)e(for)h Fm(k)i Fk(=)d Fm(n)i Fr(than)g(for)f Fm(k)i Fk(=)c Fp(b)p Fm(n=)p Fk(2)p Fp(c)p Fr(.)57 b(It)36 b(is)h(b)s(etter)g(to)f(c)m(ho)s (ose)-9 4284 y Fm(k)31 b Fk(=)d Fm(n)p Fr(.)44 b(In)34 b(this)e(case,)i(the)f(in)m(v)m(erse)h(singular)e(n)m(um)m(b)s(er)h(is) f(of)h(order)2534 4212 y Fp(p)p 2617 4212 59 4 v 72 x Fm(n)p Fr(.)44 b(W)-8 b(e)33 b(state)h(a)e(result)h(of)f([)3645 4284 y SDict begin H.S end 3645 4284 a 0 TeXcolorgray Fr(37)p 0 TeXcolorgray 3743 4220 a SDict begin H.R end 3743 4220 a 3743 4284 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Sz91) cvn H.B /ANN pdfmark end 3743 4284 a Fr(,)-9 4428 y(Theorem)g(1.2].)-9 4628 y SDict begin H.S end -9 4628 a -9 4628 a SDict begin 17.37396 H.A end -9 4628 a -9 4628 a SDict begin [/View [/XYZ H.V]/Dest (thm.13) cvn /DEST pdfmark end -9 4628 a 0 TeXcolorgray FE(Lemma)39 b(13.)p 0 TeXcolorgray 41 w Fh(L)-5 b(et)35 b Fm(n)28 b Fp(2)g Fo(N)50 b Fh(and)34 b Fm(t)28 b Fp(\025)h Fk(0)p Fh(.)44 b(Then)1284 4841 y Fo(P)1343 4731 y Fj(h)1392 4841 y Fm(s)1438 4856 y Fn(n)1501 4841 y Fk(\()p Fm(G)1616 4856 y Fn(n;n)1726 4841 y Fk(\))27 b Fp(\024)1959 4774 y Fm(t)p 1906 4819 142 4 v 1906 4838 a Fp(p)p 1989 4838 59 4 v 72 x Fm(n)2058 4731 y Fj(i)2132 4841 y Fp(\024)h Fm(t)2272 4754 y Fp(p)p 2356 4754 94 4 v 2356 4841 a Fk(2)p Fm(e:)115 5067 y Fr(This)33 b(leads)f(to)g(the)h(pro)s(of)f(of)g (Theorem)1647 5067 y SDict begin H.S end 1647 5067 a 0 TeXcolorgray Fr(3)p 0 TeXcolorgray 1696 5003 a SDict begin H.R end 1696 5003 a 1696 5067 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.3) cvn H.B /ANN pdfmark end 1696 5067 a 32 w Fr(and)1918 5067 y SDict begin H.S end 1918 5067 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 1966 5003 a SDict begin H.R end 1966 5003 a 1966 5067 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.4) cvn H.B /ANN pdfmark end 1966 5067 a 33 w Fr(as)h(presen)m(ted)h(in)e(the)h(in)m(tro)s (duction.)p 0 TeXcolorgray -9 5261 a FE(Pro)s(of)k(of)g(Theorem)902 5261 y SDict begin H.S end 902 5261 a 0 TeXcolorgray FE(3)p 0 TeXcolorgray 958 5196 a SDict begin H.R end 958 5196 a 958 5261 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.3) cvn H.B /ANN pdfmark end 958 5261 a 38 w FE(and)1212 5261 y SDict begin H.S end 1212 5261 a 0 TeXcolorgray FE(4)p 0 TeXcolorgray 1268 5196 a SDict begin H.R end 1268 5196 a 1268 5261 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.4) cvn H.B /ANN pdfmark end 1268 5261 a FE(.)p 0 TeXcolorgray 42 w Fr(T)-8 b(o)33 b(pro)m(v)m(e)i(the)f(\034rst)h (statemen)m(t,)f(let)f Fm(k)g Fk(=)c Fp(b)p Fm(n=)p Fk(2)p Fp(c)p Fr(.)48 b(W)-8 b(e)34 b(com)m(bine)-9 5405 y(Lemma)339 5405 y SDict begin H.S end 339 5405 a 0 TeXcolorgray Fr(12)p 0 TeXcolorgray 436 5341 a SDict begin H.R end 436 5341 a 436 5405 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.12) cvn H.B /ANN pdfmark end 436 5405 a 33 w Fr(and)e(Lemma)1006 5405 y SDict begin H.S end 1006 5405 a 0 TeXcolorgray Fr(11)p 0 TeXcolorgray 1104 5341 a SDict begin H.R end 1104 5341 a 1104 5405 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 1104 5405 a 32 w Fr(for)g Fm(c)c Fk(=)g(1)k Fr(with)g(Prop)s(osition)2287 5405 y SDict begin H.S end 2287 5405 a 0 TeXcolorgray Fr(10)p 0 TeXcolorgray 2384 5341 a SDict begin H.R end 2384 5341 a 2384 5405 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.10) cvn H.B /ANN pdfmark end 2384 5405 a 33 w Fr(and)g(obtain)g(that)1051 5663 y Fm(e)p Fk(\()p Fm(A)1207 5678 y Fn(n)1254 5663 y Fk(\))c Fp(\024)g Fk(78)17 b Fm(\033)1595 5678 y Fn(k)r Fl(+1)1750 5663 y Fk(+)1904 5596 y(21)p 1858 5640 191 4 v 1858 5732 a(2)1907 5660 y Fp(p)p 1990 5660 59 4 v 72 x Fm(n)2058 5523 y Fj(\022)2207 5569 y(X)2148 5785 y Fn(j)t(>)p Fi(b)p Fn(n=)p Fl(2)p Fi(c)2428 5663 y Fm(\033)2487 5622 y Fl(2)2483 5688 y Fn(j)2526 5523 y Fj(\023)2599 5545 y Fl(1)p Fn(=)p Fl(2)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.12) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(12)p 0 TeXcolorgray -9 124 a Fr(with)30 b(probabilit)m(y)f(at)i(least)f Fk(1)19 b Fp(\000)h Fm(e)1262 88 y Fi(\000)p Fn(n)1383 124 y Fp(\000)f Fm(e)1524 88 y Fi(\000)p Fn(n=)p Fl(100)1767 124 y Fr(.)43 b(The)32 b(statemen)m(t)g(follo)m(ws)d(if)h(w)m(e)i(tak)m(e)g(in)m(to)e(accoun)m (t)-9 268 y(that)1470 468 y Fm(\033)1529 427 y Fl(2)1525 493 y Fn(k)r Fl(+1)1686 468 y Fp(\024)1805 401 y Fk(4)p 1801 445 59 4 v 1801 537 a Fm(n)1929 336 y Fi(b)p Fn(n=)p Fl(2)p Fi(c)1945 374 y Fj(X)1885 590 y Fn(j)t Fl(=)p Fi(b)p Fn(n=)p Fl(4)p Fi(c)2165 468 y Fm(\033)2224 427 y Fl(2)2220 493 y Fn(j)2264 468 y Fm(:)115 730 y Fr(T)-8 b(o)29 b(pro)m(v)m(e)h(the)g(second)g(statemen)m(t,)g(w)m(e)g(set)g Fm(t)e Fk(=)f Fm(c=s)p Fr(.)42 b(W)-8 b(e)30 b(com)m(bine)e(Lemma)3015 730 y SDict begin H.S end 3015 730 a 0 TeXcolorgray Fr(13)p 0 TeXcolorgray 3113 666 a SDict begin H.R end 3113 666 a 3113 730 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.13) cvn H.B /ANN pdfmark end 3113 730 a 29 w Fr(and)h(Lemma)3673 730 y SDict begin H.S end 3673 730 a 0 TeXcolorgray Fr(11)p 0 TeXcolorgray 3770 666 a SDict begin H.R end 3770 666 a 3770 730 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.11) cvn H.B /ANN pdfmark end 3770 730 a -9 875 a Fr(with)j(Prop)s(osition)737 875 y SDict begin H.S end 737 875 a 0 TeXcolorgray Fr(10)p 0 TeXcolorgray 835 811 a SDict begin H.R end 835 811 a 835 875 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.10) cvn H.B /ANN pdfmark end 835 875 a 32 w Fr(and)h(obtain)f(that)772 1139 y Fm(e)p Fk(\()p Fm(A)928 1154 y Fn(n)976 1139 y Fk(\))27 b Fp(\024)h Fm(\033)1201 1154 y Fn(n)p Fl(+1)1361 1139 y Fk(+)1469 1072 y(1)p 1469 1116 49 4 v 1476 1208 a Fm(t)1545 969 y Fj( )1633 1072 y Fk(3)1682 1000 y Fp(p)p 1765 1000 59 4 v 72 x Fm(n)p 1633 1116 191 4 v 1704 1208 a Fk(2)1833 999 y Fj(\022)1923 1045 y(X)1930 1255 y Fn(j)t(>n)2084 1139 y Fm(\033)2143 1098 y Fl(2)2139 1164 y Fn(j)2182 999 y Fj(\023)2256 1021 y Fl(1)p Fn(=)p Fl(2)2388 1139 y Fk(+)22 b(11)p Fm(c)17 b(n)g(\033)2773 1154 y Fn(n)p Fl(+1)2909 969 y Fj(!)-9 1440 y Fr(with)40 b(probabilit)m(y)g(at)g (least)h Fk(1)28 b Fp(\000)h Fm(e)1321 1404 y Fi(\000)p Fn(c)1407 1381 y Fd(2)1441 1404 y Fn(n)1516 1440 y Fp(\000)f Fm(t)1656 1358 y Fp(p)p 1740 1358 94 4 v 1740 1440 a Fk(2)p Fm(e)p Fr(.)69 b(The)42 b(rough)f(estimates)g Fm(\033)2921 1404 y Fl(2)2917 1465 y Fn(n)p Fl(+1)3097 1440 y Fp(\024)3217 1366 y Fj(P)3322 1469 y Fn(j)t(>n)3473 1440 y Fm(\033)3532 1404 y Fl(2)3528 1465 y Fn(j)3613 1440 y Fr(and)-9 1584 y Fk(3)40 1513 y Fp(p)p 122 1513 59 4 v 122 1584 a Fm(n)q(=)p Fk(2)27 b Fp(\024)h Fk(2)p Fm(cn)k Fr(and)h Fk(1)28 b Fp(\024)g Fm(sn)k Fr(yield)g(the)h(statemen) m(t.)1742 b Fc(\003)-9 1667 y SDict begin H.S end -9 1667 a -9 1667 a SDict begin 17.37396 H.A end -9 1667 a -9 1667 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.2) cvn /DEST pdfmark end -9 1667 a 120 x Fr(3.2.)48 b FE(The)38 b(Lo)m(w)m(er)f(Bound.)49 b Fr(W)-8 b(e)33 b(w)m(an)m(t)h(to)e(giv)m(e) g(lo)m(w)m(er)h(b)s(ounds)g(on)g(the)g(radius)f(of)g(information)1044 1994 y Fp(R)1128 2009 y Fn(n)1175 1994 y Fk(\()p Fm(\033)t Fk(\))c(=)g(sup)1605 1913 y Fj(\010)1680 1994 y Fp(k)p Fm(x)p Fp(k)1835 2023 y Fl(2)1902 1994 y Fk(:)33 b Fm(x)28 b Fp(2)g(E)2192 2009 y Fn(\033)2239 1994 y Fm(;)33 b(G)2376 2009 y Fn(n)2423 1994 y Fm(x)28 b Fk(=)g(0)2659 1913 y Fj(\011)-9 2201 y Fr(whic)m(h)40 b(corresp)s(onds)h(to)e(the)h (di\036cult)m(y)g(of)f(reco)m(v)m(ering)h(an)g(unkno)m(wn)h(elemen)m(t) f Fm(x)g Fp(2)g(E)3311 2216 y Fn(\033)3397 2201 y Fr(from)f(the)-9 2345 y(information)29 b Fm(G)596 2360 y Fn(n)643 2345 y Fm(x)k Fr(in)e Fm(`)885 2360 y Fl(2)925 2345 y Fr(.)43 b(In)33 b(fact,)g(our)f(lo)m(w)m(er)h(b)s(ounds)g(already)f(hold)g(for) g(the)h(smaller)d(quan)m(tit)m(y)1039 2552 y Fp(R)1123 2511 y Fl(\()p Fn(k)r Fl(\))1123 2576 y Fn(n)1221 2552 y Fk(\()p Fm(\033)t Fk(\))e(=)f(sup)1651 2471 y Fj(\010)1726 2552 y Fp(j)o Fm(x)1808 2567 y Fn(k)1851 2552 y Fp(j)h Fk(:)33 b Fm(x)28 b Fp(2)g(E)2197 2567 y Fn(\033)2244 2552 y Fm(;)33 b(G)2381 2567 y Fn(n)2428 2552 y Fm(x)28 b Fk(=)f(0)2663 2471 y Fj(\011)-9 2759 y Fr(whic)m(h)34 b(corresp)s(onds)h(to)e(the)h(di\036cult)m(y)g(of)f(reco)m(v)m(ering)i (just)f(the)g Fm(k)s Fr(th)g(co)s(ordinate)e(of)i Fm(x)p Fr(.)47 b(Again,)33 b(w)m(e)-9 2903 y(start)f(with)g(an)h(almost)d (sure)k(estimate.)-9 3119 y SDict begin H.S end -9 3119 a -9 3119 a SDict begin 17.37396 H.A end -9 3119 a -9 3119 a SDict begin [/View [/XYZ H.V]/Dest (thm.14) cvn /DEST pdfmark end -9 3119 a 0 TeXcolorgray FE(Prop)s(osition)44 b(14.)p 0 TeXcolorgray 45 w Fh(L)-5 b(et)41 b Fm(\033)i Fp(2)c Fm(`)1209 3134 y Fl(2)1289 3119 y Fh(b)-5 b(e)41 b(non-incr)-5 b(e)g(asing.)60 b(F)-7 b(or)40 b(al)5 b(l)40 b Fm(n;)17 b(k)42 b Fp(2)d Fo(N)56 b Fh(with)41 b Fm(\033)3145 3134 y Fn(k)3226 3119 y Fp(6)p Fk(=)e(0)h Fh(we)h(have)-9 3263 y(almost)34 b(sur)-5 b(ely)760 3466 y Fp(R)844 3424 y Fl(\()p Fn(k)r Fl(\))844 3490 y Fn(n)942 3466 y Fk(\()p Fm(\033)t Fk(\))45 b Fp(\025)f Fm(\033)1298 3481 y Fn(k)1358 3295 y Fj( )1437 3466 y Fk(1)22 b Fp(\000)2019 3398 y(k)p Fk(\()p Fm(g)2154 3413 y Fn(ik)2220 3398 y Fk(\))2258 3361 y Fn(n)2258 3422 y(i)p Fl(=1)2377 3398 y Fp(k)2426 3427 y Fl(2)p 1617 3443 1251 4 v 1617 3539 a Fm(\033)1676 3498 y Fi(\000)p Fl(1)1672 3567 y Fn(k)1771 3539 y Fm(s)1817 3554 y Fn(n)1880 3459 y Fj(\000)1926 3539 y Fk(\006)1996 3555 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)2295 3459 y Fj(\001)2362 3539 y Fk(+)g Fp(k)p Fk(\()p Fm(g)2595 3554 y Fn(ik)2662 3539 y Fk(\))2700 3505 y Fn(n)2700 3565 y(i)p Fl(=1)2818 3539 y Fp(k)2878 3295 y Fj(!)2973 3466 y Fm(:)p 0 TeXcolorgray -9 3745 a Fh(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(W)d(e)45 b(ma)m(y)g(assume)h(that)f (the)g(op)s(erator)f Fm(G)1929 3760 y Fn(n)1987 3745 y Fk(:)38 b Fm(H)2133 3761 y Fg(N)p Fi(nf)p Fn(k)r Fi(g)2329 3745 y Fk(\()p Fm(\033)t Fk(\))49 b Fp(!)g Fo(R)2728 3709 y Fn(n)2826 3745 y Fr(is)c(on)m(to)g(and)g(that)g Fm(g)52 b Fk(=)-9 3889 y(\()p Fm(g)76 3904 y Fn(ik)142 3889 y Fk(\))180 3853 y Fn(n)180 3914 y(i)p Fl(=1)331 3889 y Fr(is)32 b(nonzero)h(since)g(these)g(ev)m(en)m(ts)i(o)s(ccur)e (with)f(probabilit)m(y)f(1.)43 b(Observ)m(e)34 b(that)1264 4096 y Fm(G)1341 4111 y Fn(n)1405 4016 y Fj(\000)1451 4096 y Fp(E)1513 4055 y Fg(N)p Fi(nf)p Fn(k)r Fi(g)1504 4121 y Fn(\033)1709 4016 y Fj(\001)1782 4096 y Fk(=)28 b(\006)1956 4112 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)2271 4096 y Fk(\()p Fo(B)2369 4111 y Fl(2)2414 4096 y Fk(\))17 b Fm(;)-9 4303 y Fr(where)33 b Fo(B)333 4318 y Fl(2)411 4303 y Fr(is)f(the)h(unit)f(ball)e(of)i Fm(`)1224 4318 y Fl(2)1264 4303 y Fr(.)43 b(In)33 b(particular,)e(this) h(implies)578 4510 y Fm(s)624 4525 y Fn(n)698 4510 y Fk(:=)c Fm(s)875 4525 y Fn(n)938 4429 y Fj(\000)984 4510 y Fk(\006)1054 4526 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)1353 4429 y Fj(\001)1426 4510 y Fk(=)f(sup)1693 4429 y Fj(\010)1751 4510 y Fm(\045)h Fp(\025)g Fk(0)11 b(:)33 b Fm(\045)p Fo(B)2165 4469 y Fn(n)2165 4535 y Fl(2)2246 4510 y Fp(\022)28 b Fk(\006)2421 4526 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)2736 4510 y Fk(\()p Fo(B)2834 4525 y Fl(2)2880 4510 y Fk(\))2918 4429 y Fj(\011)3004 4510 y Fm(>)f Fk(0)p Fm(:)-9 4717 y Fr(Let)32 b Fm(e)210 4681 y Fl(\()p Fn(k)r Fl(\))341 4717 y Fr(b)s(e)g(the)h Fm(k)s Fr(-th)f(standard)h(unit)f(v)m(ector)i (in)e Fm(`)1913 4732 y Fl(2)1952 4717 y Fr(.)43 b(Then)34 b(w)m(e)g(ha)m(v)m(e)1159 4924 y Fp(k)p Fm(e)1254 4883 y Fl(\()p Fn(k)r Fl(\))1352 4924 y Fp(k)1402 4939 y Fl(2)1469 4924 y Fk(=)27 b(1)97 b Fr(and)1990 4839 y Fj(\015)1990 4899 y(\015)2045 4924 y Fm(e)2090 4883 y Fl(\()p Fn(k)r Fl(\))2188 4839 y Fj(\015)2188 4899 y(\015)2243 4963 y Fn(\033)2318 4924 y Fk(=)27 b Fm(\033)2480 4883 y Fi(\000)p Fl(1)2476 4952 y Fn(k)2575 4924 y Fm(:)-9 5135 y Fr(Since)41 b(the)g(image)e(of)i Fp(E)902 5084 y Fg(N)p Fi(nf)p Fn(k)r Fi(g)893 5145 y Fn(\033)1139 5135 y Fr(under)h Fm(G)1501 5150 y Fn(n)1589 5135 y Fr(con)m(tains)f(a)f(Euclidean)h(ball)e(of)i (radius)f Fm(s)3194 5150 y Fn(n)3241 5135 y Fr(,)k(w)m(e)e(\034nd)f(an) -9 5279 y(elemen)m(t)d Fk(\026)-55 b Fm(y)36 b Fr(of)c Fp(E)608 5228 y Fg(N)p Fi(nf)p Fn(k)r Fi(g)599 5289 y Fn(\033)836 5279 y Fr(suc)m(h)i(that)1488 5460 y Fm(G)1565 5475 y Fn(n)1618 5460 y Fk(\026)-55 b Fm(y)31 b Fk(=)1804 5392 y Fm(s)1850 5407 y Fn(n)1919 5392 y Fp(\001)22 b Fm(G)2046 5407 y Fn(n)2093 5392 y Fm(e)2138 5356 y Fl(\()p Fn(k)r Fl(\))p 1804 5437 432 4 v 1817 5528 a Fp(k)p Fm(G)1944 5543 y Fn(n)1991 5528 y Fm(e)2036 5499 y Fl(\()p Fn(k)r Fl(\))2134 5528 y Fp(k)2184 5543 y Fl(2)2246 5460 y Fm(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.13) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(13)p 0 TeXcolorgray -9 124 a Fr(F)-8 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538 y Fm(z)g Fk(=)27 b(0)32 b Fr(and)h Fm(z)2242 553 y Fn(k)2313 538 y Fk(=)27 b(1)33 b Fr(as)f(w)m(ell)g(as)1025 744 y Fp(k)p Fm(z)t Fp(k)1175 773 y Fn(\033)1249 744 y Fp(\024)1354 660 y Fj(\015)1354 720 y(\015)1410 744 y Fm(e)1455 703 y Fl(\()p Fn(k)r Fl(\))1552 660 y Fj(\015)1552 720 y(\015)1608 783 y Fn(\033)1677 744 y Fk(+)22 b Fp(k)p Fm(y)t Fp(k)1926 773 y Fn(\033)2000 744 y Fp(\024)28 b Fm(\033)2164 703 y Fi(\000)p Fl(1)2160 772 y Fn(k)2281 744 y Fk(+)22 b Fm(s)2425 703 y Fi(\000)p Fl(1)2425 769 y Fn(n)2519 744 y Fp(k)p Fm(g)t Fp(k)2670 759 y Fl(2)2708 744 y Fm(:)-9 951 y Fr(The)33 b(statemen)m(t)g(is)f (obtained)g(b)m(y)1360 1184 y Fp(R)1444 1143 y Fl(\()p Fn(k)r Fl(\))1444 1208 y Fn(n)1542 1184 y Fk(\()p Fm(\033)t Fk(\))c Fp(\025)1846 1116 y(j)p Fm(z)1919 1131 y Fn(k)1962 1116 y Fp(j)p 1820 1161 197 4 v 1820 1252 a(k)p Fm(z)t Fp(k)1969 1281 y Fn(\033)2054 1184 y Fk(=)2241 1116 y(1)p 2167 1161 V 2167 1252 a Fp(k)p Fm(z)t Fp(k)2317 1281 y Fn(\033)2373 1184 y Fm(:)3693 1419 y Fc(\003)115 1613 y Fr(It)33 b(remains)f(to)g(b)s(ound)h(the)h Fm(n)p Fr(th)f(singular)e (v)-5 b(alue)32 b(of)h Fk(\006)2160 1629 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)2491 1613 y Fr(and)g(the)g(norm)f(of)g(the)i(Gaussian)-9 1758 y(v)m(ector)41 b Fk(\()p Fm(g)376 1773 y Fn(ik)442 1758 y Fk(\))480 1722 y Fn(n)480 1782 y(i)p Fl(=1)638 1758 y Fr(with)f(high)f (probabilit)m(y)-8 b(.)64 b(F)-8 b(or)39 b(b)s(oth)h(estimates,)i(w)m (e)f(use)g(the)f(follo)m(wing)d(concen-)-9 1902 y(tration)k(result)i (for)f(c)m(hi-square)h(random)f(v)-5 b(ariables)42 b(going)f(bac)m(k)j (to)f(Lauren)m(t)g(and)g(Massart)g([)3645 1902 y SDict begin H.S end 3645 1902 a 0 TeXcolorgray Fr(20)p 0 TeXcolorgray 3743 1838 a SDict begin H.R end 3743 1838 a 3743 1902 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.LM00) cvn H.B /ANN pdfmark end 3743 1902 a Fr(,)-9 2046 y(Lemma)30 b(1].)43 b(Alternativ)m(ely)-8 b(,)31 b(one)h(could)f(use)i(the)f (concen)m(tration)f(of)h(Gaussian)f(random)g(v)m(ectors)i(in)-9 2190 y(Banac)m(h)g(spaces)h(\(see,)f(e.g.,)g([)1093 2190 y SDict begin H.S end 1093 2190 a 0 TeXcolorgray Fr(21)p 0 TeXcolorgray 1191 2126 a SDict begin H.R end 1191 2126 a 1191 2190 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Ledoux2001) cvn H.B /ANN pdfmark end 1191 2190 a Fr(,)g(Prop)s(osition)d (2.18]\).)-9 2407 y SDict begin H.S end -9 2407 a -9 2407 a SDict begin 17.37396 H.A end -9 2407 a -9 2407 a SDict begin [/View [/XYZ H.V]/Dest (thm.15) cvn /DEST pdfmark end -9 2407 a 0 TeXcolorgray FE(Lemma)52 b(15.)p 0 TeXcolorgray 47 w Fh(F)-7 b(or)45 b Fk(1)i Fp(\024)h Fm(j)53 b Fp(\024)48 b Fm(m)p Fh(,)h(let)c Fm(u)1590 2422 y Fn(j)1672 2407 y Fh(b)-5 b(e)45 b(indep)-5 b(endent)44 b(c)-5 b(enter)g(e)g(d)45 b(Gaussian)g(variables)f(with)-9 2551 y(varianc)-5 b(e)33 b Fm(a)431 2566 y Fn(j)468 2551 y Fh(.)45 b(Then,)34 b(for)g(any)h Fm(\016)d Fp(2)c Fk(\(0)p Fm(;)17 b Fk(1])p Fh(,)34 b(we)g(have)836 2813 y Fo(P)895 2672 y Fj(\024)1007 2688 y Fn(m)966 2718 y Fj(X)977 2928 y Fn(j)t Fl(=1)1127 2813 y Fm(u)1183 2772 y Fl(2)1183 2838 y Fn(j)1266 2813 y Fp(\024)45 b Fk(\(1)22 b Fp(\000)g Fm(\016)t Fk(\))1738 2688 y Fn(m)1698 2718 y Fj(X)1708 2928 y Fn(j)t Fl(=1)1858 2813 y Fm(a)1909 2828 y Fn(j)1946 2672 y Fj(\025)2026 2813 y Fp(\024)28 b Fk(exp)2297 2672 y Fj(\022)2370 2813 y Fp(\000)2457 2745 y Fm(\016)2504 2709 y Fl(2)2561 2745 y Fp(k)p Fm(a)p Fp(k)2712 2774 y Fl(1)p 2457 2790 294 4 v 2458 2881 a Fk(4)17 b Fp(k)p Fm(a)p Fp(k)2675 2910 y Fi(1)2761 2672 y Fj(\023)2851 2813 y Fm(;)838 3159 y Fo(P)897 3018 y Fj(\024)1009 3034 y Fn(m)968 3064 y Fj(X)979 3274 y Fn(j)t Fl(=1)1128 3159 y Fm(u)1184 3117 y Fl(2)1184 3183 y Fn(j)1268 3159 y Fp(\025)44 b Fk(\(1)22 b(+)g Fm(\016)t Fk(\))1738 3034 y Fn(m)1698 3064 y Fj(X)1708 3274 y Fn(j)t Fl(=1)1858 3159 y Fm(a)1909 3174 y Fn(j)1946 3018 y Fj(\025)2026 3159 y Fp(\024)28 b Fk(exp)2297 3018 y Fj(\022)2370 3159 y Fp(\000)2481 3091 y Fm(\016)2528 3054 y Fl(2)2584 3091 y Fp(k)p Fm(a)p Fp(k)2735 3120 y Fl(1)p 2457 3136 340 4 v 2457 3227 a Fk(16)17 b Fp(k)p Fm(a)p Fp(k)2723 3256 y Fi(1)2807 3018 y Fj(\023)2897 3159 y Fm(:)p 0 TeXcolorgray -9 3452 a Fh(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(The)33 b(lemma)d([)830 3452 y SDict begin H.S end 830 3452 a 0 TeXcolorgray Fr(20)p 0 TeXcolorgray 928 3388 a SDict begin H.R end 928 3388 a 928 3452 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.LM00) cvn H.B /ANN pdfmark end 928 3452 a Fr(,)j(Lemma)e(1])h(states)h(that,)g(for)f(all)e Fm(t)e(>)g Fk(0)p Fr(,)k(w)m(e)i(ha)m(v)m(e)867 3722 y Fo(P)943 3552 y Fj(")1043 3598 y Fn(m)1003 3627 y Fj(X)1013 3837 y Fn(j)t Fl(=1)1163 3722 y Fm(u)1219 3681 y Fl(2)1219 3747 y Fn(j)1286 3722 y Fp(\024)28 b(k)p Fm(a)p Fp(k)1542 3751 y Fl(1)1603 3722 y Fp(\000)23 b Fk(2)17 b Fp(k)o Fm(a)p Fp(k)1919 3751 y Fl(2)1975 3722 y Fm(t)2010 3552 y Fj(#)2096 3722 y Fp(\024)28 b Fm(e)2246 3681 y Fi(\000)p Fn(t)2326 3657 y Fd(2)2366 3722 y Fm(;)867 4079 y Fo(P)943 3909 y Fj(")1043 3954 y Fn(m)1003 3984 y Fj(X)1013 4194 y Fn(j)t Fl(=1)1163 4079 y Fm(u)1219 4038 y Fl(2)1219 4104 y Fn(j)1286 4079 y Fp(\025)g(k)p Fm(a)p Fp(k)1542 4108 y Fl(1)1603 4079 y Fk(+)22 b(2)17 b Fp(k)p Fm(a)p Fp(k)1918 4108 y Fl(2)1974 4079 y Fm(t)22 b Fk(+)g(2)17 b Fp(k)o Fm(a)p Fp(k)2345 4108 y Fi(1)2437 4079 y Fm(t)2472 4038 y Fl(2)2511 3909 y Fj(#)2597 4079 y Fp(\024)28 b Fm(e)2747 4038 y Fi(\000)p Fn(t)2827 4014 y Fd(2)2866 4079 y Fm(:)-9 4357 y Fr(The)33 b(form)m(ulation)d(of)i(Lemma)1176 4357 y SDict begin H.S end 1176 4357 a 0 TeXcolorgray Fr(15)p 0 TeXcolorgray 1273 4292 a SDict begin H.R end 1273 4292 a 1273 4357 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.15) cvn H.B /ANN pdfmark end 1273 4357 a 33 w Fr(follo)m(ws)f(if)g(w)m(e)j(put)553 4632 y Fm(t)28 b Fk(=)730 4565 y Fm(\016)21 b Fp(k)p Fm(a)p Fp(k)945 4594 y Fl(1)p 729 4609 256 4 v 729 4701 a Fk(2)c Fp(k)p Fm(a)p Fp(k)946 4730 y Fl(2)995 4632 y Fm(;)212 b Fr(resp)s(ectiv)m(ely)196 b Fm(t)28 b Fk(=)f(min)2274 4462 y Fj(\()2365 4565 y Fm(\016)20 b Fp(k)p Fm(a)p Fp(k)2579 4594 y Fl(1)p 2364 4609 V 2364 4701 a Fk(4)d Fp(k)o Fm(a)p Fp(k)2580 4730 y Fl(2)2629 4632 y Fm(;)2673 4455 y Fj(s)p 2773 4455 311 4 v 2801 4565 a Fm(\016)k Fp(k)p Fm(a)p Fp(k)3016 4594 y Fl(1)p 2783 4609 291 4 v 2783 4701 a Fk(4)c Fp(k)o Fm(a)p Fp(k)2999 4730 y Fi(1)3084 4462 y Fj(\))3180 4632 y Fm(:)-9 4907 y Fr(The)33 b(desired)g(probabilit)m(y)e(estimate)g (then)i(follo)m(ws)f(b)m(y)h(using)f Fp(k)p Fm(a)p Fp(k)2495 4871 y Fl(2)2495 4931 y(2)2562 4907 y Fp(\024)c(k)p Fm(a)p Fp(k)2818 4922 y Fl(1)2858 4907 y Fp(k)p Fm(a)p Fp(k)3009 4922 y Fi(1)3083 4907 y Fr(.)583 b Fc(\003)115 5101 y Fr(In)39 b(particular,)g(the)g(norm)f(of)g(the)h(Gaussian)f(v)m(ector)i Fk(\()p Fm(g)2263 5116 y Fn(ik)2329 5101 y Fk(\))2367 5065 y Fn(n)2367 5126 y(i)p Fl(=1)2524 5101 y Fr(concen)m(trates)h (around)3429 5029 y Fp(p)p 3512 5029 59 4 v 72 x Fm(n)p Fr(.)62 b(T)-8 b(o)-9 5245 y(b)s(ound)37 b(the)g Fm(n)p Fr(th)h(singular)d(v)-5 b(alue)37 b(of)f Fk(\006)1471 5261 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)1807 5245 y Fr(w)m(e)i(shall)e(use)i(Gordon's)f(min-max)d (theorem.)57 b(Let)-9 5389 y(us)33 b(state)g(Gordon's)f(theorem)g([) 1179 5389 y SDict begin H.S end 1179 5389 a 0 TeXcolorgray Fr(11)p 0 TeXcolorgray 1277 5325 a SDict begin H.R end 1277 5325 a 1277 5389 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.Go88) cvn H.B /ANN pdfmark end 1277 5389 a Fr(,)h(Lemma)e(3.1])h (in)g(a)g(form)f(whic)m(h)i(can)g(b)s(e)g(found)f(in)g([)3299 5389 y SDict begin H.S end 3299 5389 a 0 TeXcolorgray Fr(39)p 0 TeXcolorgray 3397 5325 a SDict begin H.R end 3397 5325 a 3397 5389 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (cite.HOT15) cvn H.B /ANN pdfmark end 3397 5389 a Fr(].)-9 5606 y SDict begin H.S end -9 5606 a -9 5606 a SDict begin 17.37396 H.A end -9 5606 a -9 5606 a SDict begin [/View [/XYZ H.V]/Dest (thm.16) cvn /DEST pdfmark end -9 5606 a 0 TeXcolorgray FE(Lemma)44 b(16)e Fr(\(Gordon's)37 b(min-max)e(theorem\))p FE(.)p 0 TeXcolorgray 43 w Fh(L)-5 b(et)40 b Fm(n;)17 b(m)35 b Fp(2)h Fo(N)54 b Fh(and)39 b(let)g Fm(S)2884 5621 y Fl(1)2958 5606 y Fp(\022)d Fo(R)3137 5570 y Fn(n)3190 5606 y Fh(,)k Fm(S)3320 5621 y Fl(2)3395 5606 y Fp(\022)c Fo(R)3574 5570 y Fn(m)3685 5606 y Fh(b)-5 b(e)-9 5750 y(c)g(omp)g(act)40 b(sets.)63 b(Assume)40 b(that)i Fm( )15 b Fk(:)35 b Fm(S)1398 5765 y Fl(1)1464 5750 y Fp(\002)28 b Fm(S)1629 5765 y Fl(2)1707 5750 y Fp(!)39 b Fo(R)52 b Fh(is)40 b(a)h(c)-5 b(ontinuous)41 b(mapping.)61 b(L)-5 b(et)42 b Fm(G)d Fp(2)g Fo(R)3570 5714 y Fn(m)p Fi(\002)p Fn(n)3740 5750 y Fh(,)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.14) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(14)p 0 TeXcolorgray -9 121 a Fm(u)28 b Fp(2)h Fo(R)236 85 y Fn(m)309 121 y Fh(,)35 b(and)g Fm(v)e Fp(2)c Fo(R)805 85 y Fn(n)894 121 y Fh(b)-5 b(e)35 b(indep)-5 b(endent)34 b(r)-5 b(andom)34 b(obje)-5 b(cts)35 b(with)h(indep)-5 b(endent)34 b(standar)-5 b(d)34 b(Gaussian)-9 265 y(entries.)44 b(Mor)-5 b(e)g(over,)34 b(de\034ne)702 472 y Fk(\010)772 487 y Fl(1)812 472 y Fk(\()p Fm(G)p Fk(\))82 b(:=)i(min)1233 534 y Fn(x)p Fi(2)p Fn(S)1363 543 y Fd(1)1414 472 y Fk(max)1424 534 y Fn(y)r Fi(2)p Fn(S)1551 543 y Fd(2)1612 361 y Fj(\020)1672 472 y Fp(h)p Fm(y)t(;)17 b(Gx)p Fp(i)k Fk(+)h Fm( )t Fk(\()p Fm(x;)17 b(y)t Fk(\))2391 361 y Fj(\021)2449 472 y Fm(;)629 680 y Fk(\010)699 695 y Fl(2)739 680 y Fk(\()p Fm(u;)g(v)t Fk(\))81 b(:=)j(min)1233 742 y Fn(x)p Fi(2)p Fn(S)1363 751 y Fd(1)1414 680 y Fk(max)1424 742 y Fn(y)r Fi(2)p Fn(S)1551 751 y Fd(2)1612 569 y Fj(\020)1672 680 y Fp(k)p Fm(x)p Fp(k)1827 695 y Fl(2)1866 680 y Fp(h)p Fm(u;)17 b(y)t Fp(i)j Fk(+)i Fp(k)p Fm(y)t Fp(k)2366 695 y Fl(2)2405 680 y Fp(h)p Fm(v)t(;)17 b(x)p Fp(i)k Fk(+)h Fm( )t Fk(\()p Fm(x;)17 b(y)t Fk(\))3046 569 y Fj(\021)3105 680 y Fm(:)-9 904 y Fh(Then,)33 b(for)i(al)5 b(l)34 b Fm(c)28 b Fp(2)g Fo(R)5 b Fh(,)41 b(we)34 b(have)1151 1111 y Fo(P)1210 1030 y Fj(\002)1253 1111 y Fk(\010)1323 1126 y Fl(1)1363 1111 y Fk(\()p Fm(G)p Fk(\))27 b Fm(<)h(c)1689 1030 y Fj(\003)1758 1111 y Fp(\024)g Fk(2)17 b Fo(P)1988 1030 y Fj(\002)2031 1111 y Fk(\010)2101 1126 y Fl(2)2141 1111 y Fk(\()p Fm(u;)g(v)t Fk(\))26 b Fp(\024)j Fm(c)2542 1030 y Fj(\003)2583 1111 y Fm(:)115 1327 y Fr(This)47 b(yields)g(the)g(follo)m(wing)d(lo)m(w)m(er)k(b)s(ound)f(on)g(the)g (smallest)e(singular)h(v)-5 b(alue)46 b(of)h(structured)-9 1472 y(Gaussian)32 b(matrices.)42 b(Note)33 b(that)f(this)g(is)g(a)h (generalization)d(of)i(Lemma)2737 1472 y SDict begin H.S end 2737 1472 a 0 TeXcolorgray Fr(12)p 0 TeXcolorgray 2834 1407 a SDict begin H.R end 2834 1407 a 2834 1472 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.12) cvn H.B /ANN pdfmark end 2834 1472 a Fr(.)-9 1688 y SDict begin H.S end -9 1688 a -9 1688 a SDict begin 17.37396 H.A end -9 1688 a -9 1688 a SDict begin [/View [/XYZ H.V]/Dest (thm.17) cvn /DEST pdfmark end -9 1688 a 0 TeXcolorgray FE(Lemma)39 b(17.)p 0 TeXcolorgray 42 w Fh(L)-5 b(et)35 b Fm(A)29 b Fp(2)f Fo(R)1003 1652 y Fn(m)q Fi(\002)p Fn(n)1209 1688 y Fh(b)-5 b(e)34 b(a)h(r)-5 b(andom)34 b(matrix)h(whose)f(entries)h Fm(a)2739 1703 y Fn(ij)2835 1688 y Fh(ar)-5 b(e)35 b(c)-5 b(enter)g(e)g(d)34 b(Gaussian)-9 1832 y(variables)f(with)i(varianc)-5 b(e)34 b Fm(a)1048 1847 y Fn(i)1111 1832 y Fh(for)g(al)5 b(l)35 b Fm(i)28 b Fp(\024)g Fm(m)35 b Fh(and)f Fm(j)g Fp(\024)28 b Fm(n)p Fh(.)45 b(Then,)34 b(for)h(al)5 b(l)34 b Fm(\016)e Fp(2)c Fk(\(0)p Fm(;)17 b Fk(1\))p Fh(,)34 b(we)g(have)-9 2074 y Fr(\(3\))236 2074 y SDict begin H.S end 236 2074 a 236 2074 a SDict begin 17.37396 H.A end 236 2074 a 236 2074 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.3) cvn /DEST pdfmark end 236 2074 a Fo(P)312 1934 y Fj(\024)366 2074 y Fm(s)412 2089 y Fn(n)459 2074 y Fk(\()p Fm(A)p Fk(\))28 b Fp(\024)741 1957 y Fj(q)p 841 1957 501 4 v 117 x Fk(\(1)22 b Fp(\000)g Fm(\016)t Fk(\))17 b Fp(k)p Fm(a)p Fp(k)1302 2104 y Fl(1)1363 2074 y Fp(\000)1463 1957 y Fj(q)p 1562 1957 593 4 v 1562 2074 a Fk(\(1)22 b(+)g Fm(\016)t Fk(\))p Fm(n)17 b Fp(k)p Fm(a)p Fp(k)2080 2104 y Fi(1)2155 1934 y Fj(\025)2235 2074 y Fp(\024)28 b Fk(4)17 b(exp)2571 1934 y Fj(\022)2644 2074 y Fp(\000)2737 2007 y Fm(\016)2784 1971 y Fl(2)p 2731 2052 98 4 v 2731 2143 a Fk(16)2856 2074 y(min)3035 1934 y Fj(\032)3110 2074 y Fm(n;)3239 2007 y Fp(k)p Fm(a)p Fp(k)3390 2036 y Fl(1)p 3222 2052 226 4 v 3222 2143 a Fp(k)o Fm(a)p Fp(k)3372 2172 y Fi(1)3457 1934 y Fj(\033\023)3622 2074 y Fm(:)p 0 TeXcolorgray -9 2321 a Fh(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(Note)25 b(that)f(the)i(statemen)m(t)f(is)g(trivial)d(if)i Fm(m)k Fp(\024)g Fm(n)p Fr(.)41 b(W)-8 b(e)25 b(ma)m(y)g(assume)g(that)g(the)g Fm(a)3233 2336 y Fn(i)3287 2321 y Fr(are)g(p)s(ositiv)m(e)-9 2465 y(since)38 b(an)g(additional)d(ro)m(w)j(of)g(zeros)g(do)s(es)h (neither)f(c)m(hange)h Fm(s)2336 2480 y Fn(n)2382 2465 y Fk(\()p Fm(A)p Fk(\))f Fr(nor)g(the)h(norms)e(of)h(the)g(v)m(ector)-9 2609 y Fm(a)p Fr(.)43 b(W)-8 b(e)32 b(ha)m(v)m(e)i(the)e(iden)m(tit)m (y)g Fm(A)c Fk(=)f Fm(D)s(G)32 b Fr(where)h Fm(G)28 b Fp(2)g Fo(R)1974 2573 y Fn(m)p Fi(\002)p Fn(n)2176 2609 y Fr(is)k(a)g(random)f(matrix)f(with)i(indep)s(enden)m(t)-9 2754 y(standard)g(Gaussian)g(en)m(tries)h(and)g Fm(D)e Fp(2)d Fo(R)1590 2717 y Fn(m)p Fi(\002)p Fn(m)1812 2754 y Fr(is)k(the)h(diagonal)d(matrix)1318 2961 y Fm(D)h Fk(=)c(diag)17 b(\()1768 2891 y Fp(p)p 1851 2891 91 4 v 70 x Fm(a)1902 2976 y Fl(1)1942 2961 y Fm(;)g(:)g(:)g(:)e(;)2160 2891 y Fp(p)p 2243 2891 118 4 v 70 x Fm(a)2294 2976 y Fn(m)2361 2961 y Fk(\))h Fm(:)-9 3167 y Fr(W)-8 b(e)32 b(w)m(an)m(t)i(to)e(apply)g(Gordon's)g(theorem)g(for)g(the)h(matrix)d Fm(G)j Fr(and)f Fm( )g Fk(=)27 b(0)p Fr(,)33 b(where)g Fm(S)3161 3182 y Fl(1)3233 3167 y Fr(is)f(the)h(sphere)-9 3312 y(in)e Fo(R)170 3275 y Fn(n)256 3312 y Fr(and)h Fm(S)505 3327 y Fl(2)577 3312 y Fr(is)g(the)h(image)e(of)h(the)h (sphere)h(in)e Fo(R)1888 3275 y Fn(m)1993 3312 y Fr(under)h Fm(D)s Fr(.)43 b(Then)34 b(w)m(e)g(ha)m(v)m(e)74 3555 y Fk(\010)144 3570 y Fl(1)183 3555 y Fk(\()p Fm(G)p Fk(\))28 b(=)g(min)468 3617 y Fn(x)p Fi(2)p Fn(S)598 3626 y Fd(1)648 3555 y Fk(max)658 3617 y Fn(y)r Fi(2)p Fn(S)785 3626 y Fd(2)830 3555 y Fp(h)p Fm(y)t(;)17 b(Gx)p Fp(i)26 b Fk(=)64 b(min)1266 3621 y Fi(k)p Fn(x)p Fi(k)1376 3642 y Fd(2)1411 3621 y Fl(=1)1542 3555 y Fk(max)1518 3621 y Fi(k)p Fn(z)s Fi(k)1624 3642 y Fd(2)1658 3621 y Fl(=1)1748 3555 y Fp(h)p Fm(D)s(z)t(;)17 b(Gx)p Fp(i)1751 3785 y Fk(=)63 b(min)1854 3851 y Fi(k)p Fn(x)p Fi(k)1964 3872 y Fd(2)1999 3851 y Fl(=1)2131 3785 y Fk(max)2106 3851 y Fi(k)p Fn(z)s Fi(k)2212 3872 y Fd(2)2247 3851 y Fl(=1)2337 3785 y Fp(h)p Fm(z)t(;)17 b(Ax)p Fp(i)28 b Fk(=)64 b(min)2768 3851 y Fi(k)p Fn(x)p Fi(k)2878 3872 y Fd(2)2912 3851 y Fl(=1)3019 3785 y Fp(k)p Fm(Ax)p Fp(k)3247 3814 y Fl(2)3314 3785 y Fk(=)28 b Fm(s)3464 3800 y Fn(n)3511 3785 y Fk(\()p Fm(A)p Fk(\))p Fm(:)-9 4021 y Fr(On)36 b(the)h(other)g(hand,)h(if)e Fm(u)e Fp(2)i Fo(R)1216 3985 y Fn(n)1305 4021 y Fr(and)h Fm(v)i Fp(2)c Fo(R)1752 3985 y Fn(m)1861 4021 y Fr(are)i(standard)g (Gaussian)f(v)m(ectors,)j(the)e(c)m(hoice)h(of)-9 4166 y Fm(z)32 b Fk(=)c Fm(D)s(u=)p Fp(k)p Fm(D)s(u)p Fp(k)601 4181 y Fl(2)670 4166 y Fr(yields)74 4412 y Fk(\010)144 4427 y Fl(2)183 4412 y Fk(\()p Fm(u;)17 b(v)t Fk(\))27 b(=)h(min)541 4474 y Fn(x)p Fi(2)p Fn(S)671 4483 y Fd(1)721 4412 y Fk(max)731 4474 y Fn(y)r Fi(2)p Fn(S)858 4483 y Fd(2)920 4301 y Fj(\020)996 4412 y Fp(h)o Fm(u;)17 b(y)t Fp(i)k Fk(+)h Fp(k)o Fm(y)t Fp(k)1494 4441 y Fl(2)1550 4412 y Fp(h)p Fm(v)t(;)17 b(x)p Fp(i)1794 4301 y Fj(\021)1043 4656 y Fk(=)64 b(min)1147 4722 y Fi(k)p Fn(x)p Fi(k)1257 4743 y Fd(2)1292 4722 y Fl(=1)1423 4656 y Fk(max)1398 4722 y Fi(k)q Fn(z)s Fi(k)1505 4743 y Fd(2)1539 4722 y Fl(=1)1646 4545 y Fj(\020)1722 4656 y Fp(h)p Fm(u;)17 b(D)s(z)t Fp(i)k Fk(+)h Fp(k)p Fm(D)s(z)t Fp(k)2386 4685 y Fl(2)2442 4656 y Fp(h)o Fm(v)t(;)17 b(x)p Fp(i)2686 4545 y Fj(\021)1150 4944 y Fp(\025)64 b Fk(min)1255 5011 y Fi(k)p Fn(x)p Fi(k)1365 5031 y Fd(2)1400 5011 y Fl(=1)1506 4834 y Fj(\020)1583 4944 y Fp(k)o Fm(D)s(u)p Fp(k)1821 4973 y Fl(2)1883 4944 y Fk(+)1991 4876 y Fp(k)p Fm(D)2125 4840 y Fl(2)2164 4876 y Fm(u)p Fp(k)2269 4905 y Fl(2)p 1991 4921 318 4 v 2011 5013 a Fp(k)o Fm(D)s(u)p Fp(k)2250 5042 y Fl(2)2335 4944 y Fp(h)p Fm(v)t(;)17 b(x)p Fp(i)2579 4834 y Fj(\021)1531 5233 y Fk(=)28 b Fp(k)p Fm(D)s(u)p Fp(k)1874 5262 y Fl(2)1935 5233 y Fp(\000)2045 5165 y(k)p Fm(D)2179 5129 y Fl(2)2218 5165 y Fm(u)p Fp(k)2323 5194 y Fl(2)p 2045 5210 V 2065 5301 a Fp(k)o Fm(D)s(u)p Fp(k)2303 5330 y Fl(2)2389 5233 y Fp(k)p Fm(v)t Fp(k)2539 5262 y Fl(2)2607 5233 y Fp(\025)g(k)p Fm(D)s(u)p Fp(k)2951 5262 y Fl(2)3012 5233 y Fp(\000)3112 5115 y Fj(q)p 3211 5115 226 4 v 3211 5233 a Fp(k)p Fm(a)p Fp(k)3362 5262 y Fi(1)3454 5233 y Fp(k)o Fm(v)t Fp(k)3604 5262 y Fl(2)3660 5233 y Fm(:)-9 5470 y Fr(Theorem)402 5470 y SDict begin H.S end 402 5470 a 0 TeXcolorgray Fr(16)p 0 TeXcolorgray 500 5406 a SDict begin H.R end 500 5406 a 500 5470 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.16) cvn H.B /ANN pdfmark end 500 5470 a 32 w Fr(implies)i(for)i(all)f Fm(c)d Fp(2)g Fo(R)43 b Fr(that)428 5686 y Fo(P)487 5575 y Fj(h)536 5686 y Fm(s)582 5701 y Fn(n)629 5686 y Fk(\()p Fm(A)p Fk(\))28 b Fm(<)f(c)951 5575 y Fj(i)1026 5686 y Fp(\024)h Fk(2)p Fo(P)1239 5575 y Fj(h)1288 5686 y Fk(\010)1358 5701 y Fl(2)1398 5686 y Fk(\()p Fm(u;)17 b(v)t Fk(\))26 b Fp(\024)j Fm(c)1799 5575 y Fj(i)1873 5686 y Fp(\024)f Fk(2)p Fo(P)2086 5575 y Fj(h)2152 5686 y Fp(k)p Fm(D)s(u)p Fp(k)2391 5715 y Fl(2)2452 5686 y Fp(\000)2552 5568 y Fj(q)p 2651 5568 226 4 v 2651 5686 a Fp(k)p Fm(a)p Fp(k)2802 5715 y Fi(1)2894 5686 y Fp(k)o Fm(v)t Fp(k)3044 5715 y Fl(2)3111 5686 y Fp(\024)g Fm(c)3258 5575 y Fj(i)3305 5686 y Fm(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 15 15 TeXDict begin HPSdict begin 15 14 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.15) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(15)p 0 TeXcolorgray -9 131 a Fr(T)-8 b(o)35 b(obtain)f(the)i(statemen)m(t)f(of)g(our)g(lemma,)f(w)m(e)i(set)g Fm(c)c Fk(=)2187 46 y Fj(p)p 2287 46 484 4 v 85 x Fk(\(1)22 b Fp(\000)g Fm(\016)t Fk(\))p Fp(k)p Fm(a)p Fp(k)2731 146 y Fl(1)2794 131 y Fp(\000)2896 46 y Fj(p)p 2996 46 576 4 v 85 x Fk(\(1)f(+)h Fm(\016)t Fk(\))p Fm(n)p Fp(k)p Fm(a)p Fp(k)3496 146 y Fi(1)3571 131 y Fr(.)52 b(By)-9 275 y(Lemma)339 275 y SDict begin H.S end 339 275 a 0 TeXcolorgray Fr(15)p 0 TeXcolorgray 436 211 a SDict begin H.R end 436 211 a 436 275 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.15) cvn H.B /ANN pdfmark end 436 275 a Fr(,)33 b(w)m(e)g(ha)m(v)m(e)838 512 y Fo(P)897 402 y Fj(h)962 512 y Fp(k)p Fm(D)s(u)p Fp(k)1201 541 y Fl(2)1285 512 y Fp(\024)1407 395 y Fj(q)p 1506 395 501 4 v 1506 512 a Fk(\(1)22 b Fp(\000)h Fm(\016)t Fk(\))17 b Fp(k)o Fm(a)p Fp(k)1967 541 y Fl(1)2007 402 y Fj(i)2098 512 y Fp(\024)45 b Fk(exp)2385 372 y Fj(\022)2459 512 y Fp(\000)2546 444 y Fm(\016)2593 408 y Fl(2)2649 444 y Fp(k)p Fm(a)p Fp(k)2800 474 y Fl(1)p 2546 489 294 4 v 2547 581 a Fk(4)17 b Fp(k)p Fm(a)p Fp(k)2764 610 y Fi(1)2849 372 y Fj(\023)-9 749 y Fr(and)1010 904 y Fo(P)1069 793 y Fj(h)1134 904 y Fp(k)p Fm(v)t Fp(k)1285 933 y Fl(2)1368 904 y Fp(\025)1490 814 y Fj(p)p 1590 814 350 4 v 90 x Fk(\(1)22 b(+)g Fm(\016)t Fk(\))p Fm(n)1940 793 y Fj(i)2031 904 y Fp(\024)45 b Fk(exp)2318 764 y Fj(\022)2392 904 y Fp(\000)2479 837 y Fm(\016)2526 800 y Fl(2)2566 837 y Fm(n)p 2479 881 145 4 v 2503 972 a Fk(16)2634 764 y Fj(\023)2724 904 y Fm(:)-9 1107 y Fr(No)m(w)33 b(the)g(statemen)m(t)g(is)f(obtained)g (from)f(a)h(union)g(b)s(ound.)1483 b Fc(\003)115 1301 y Fr(W)-8 b(e)34 b(need)h(the)f(statemen)m(t)g(of)f(Lemma)1597 1301 y SDict begin H.S end 1597 1301 a 0 TeXcolorgray Fr(17)p 0 TeXcolorgray 1695 1237 a SDict begin H.R end 1695 1237 a 1695 1301 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.17) cvn H.B /ANN pdfmark end 1695 1301 a 34 w Fr(for)g(matrices)f(with)i(in\034nitely)e(man)m(y)h(ro)m(ws,)i(whic)m (h)g(is)-9 1445 y(obtained)c(from)h(a)g(simple)f(limit)e(argumen)m(t.) -9 1662 y SDict begin H.S end -9 1662 a -9 1662 a SDict begin 17.37396 H.A end -9 1662 a -9 1662 a SDict begin [/View [/XYZ H.V]/Dest (thm.18) cvn /DEST pdfmark end -9 1662 a 0 TeXcolorgray FE(Lemma)39 b(18.)p 0 TeXcolorgray 41 w Fh(F)-7 b(ormula)41 b Fr(\()1006 1662 y SDict begin H.S end 1006 1662 a 0 TeXcolorgray Fr(3)p 0 TeXcolorgray 1055 1597 a SDict begin H.R end 1055 1597 a 1055 1662 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.3.3) cvn H.B /ANN pdfmark end 1055 1662 a Fr(\))35 b Fh(also)f(holds)g(for)h Fm(m)28 b Fk(=)f Fp(1)35 b Fh(pr)-5 b(ovide)g(d)33 b(that)j Fm(a)27 b Fp(2)h Fm(`)2880 1677 y Fl(1)2920 1662 y Fh(.)p 0 TeXcolorgray -9 1878 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(Again,)36 b(w)m(e)h(ma)m(y)f(assume)g(that)g Fm(a)g Fr(is)g(strictly)f(p)s(ositiv)m(e.)53 b(F)-8 b(or)36 b Fm(m)e Fp(2)g Fo(N)51 b Fr(let)35 b Fm(A)3180 1893 y Fn(m)3283 1878 y Fr(b)s(e)h(the)h(sub-)-9 2022 y(matrix)29 b(consisting)g(of)h(the)h(\034rst)g Fm(m)g Fr(ro)m(ws)h(of)e Fm(A)g Fr(and)h(let)f Fm(a)2158 1986 y Fl(\()p Fn(m)p Fl(\))2310 2022 y Fr(b)s(e)h(the)g(sub-v)m(ector)h(consisting)e(of)g (the)-9 2166 y(\034rst)j Fm(m)f Fr(en)m(tries)h(of)g Fm(a)p Fr(.)43 b(W)-8 b(e)33 b(use)h(the)f(notation)877 2373 y Fm(c)919 2388 y Fn(m)986 2373 y Fk(\()p Fm(\016)t Fk(\))27 b(=)1240 2251 y Fj(q)p 1340 2251 606 4 v 122 x Fk(\(1)21 b Fp(\000)i Fm(\016)t Fk(\))p Fp(k)p Fm(a)1734 2345 y Fl(\()p Fn(m)p Fl(\))1855 2373 y Fp(k)1905 2388 y Fl(1)1967 2373 y Fp(\000)2066 2251 y Fj(q)p 2166 2251 697 4 v 122 x Fk(\(1)f(+)g Fm(\016)t Fk(\))p Fm(n)p Fp(k)p Fm(a)2617 2345 y Fl(\()p Fn(m)p Fl(\))2739 2373 y Fp(k)2789 2388 y Fi(1)2863 2373 y Fm(;)870 2645 y(p)919 2660 y Fn(m)986 2645 y Fk(\()p Fm(\016)t Fk(\))27 b(=)h(4)17 b(exp)1471 2475 y Fj( )1550 2645 y Fp(\000)1643 2578 y Fm(\016)1690 2542 y Fl(2)p 1637 2623 98 4 v 1637 2714 a Fk(16)1761 2645 y(min)1941 2475 y Fj(\()2021 2645 y Fm(n;)2145 2483 y Fj(\015)2145 2543 y(\015)2200 2568 y Fm(a)2251 2531 y Fl(\()p Fn(m)p Fl(\))2373 2483 y Fj(\015)2373 2543 y(\015)2428 2607 y Fl(1)p 2133 2623 348 4 v 2133 2714 a Fp(k)p Fm(a)2234 2685 y Fl(\()p Fn(m)p Fl(\))2355 2714 y Fp(k)2405 2743 y Fi(1)2490 2475 y Fj(\)!)2666 2645 y Fm(;)-9 2916 y Fr(where)35 b Fm(c)p Fk(\()p Fm(\016)t Fk(\))f Fr(and)g Fm(p)p Fk(\()p Fm(\016)t Fk(\))g Fr(corresp)s(ond)h (to)f(the)h(case)g Fm(m)c Fk(=)g Fp(1)p Fr(.)48 b(F)-8 b(or)33 b(an)m(y)i Fm(")c(>)f Fk(0)k Fr(with)g Fm(")c(<)h(\016)t(=)p Fk(2)i Fr(w)m(e)j(can)-9 3060 y(c)m(ho)s(ose)e Fm(m)c Fp(\025)g Fm(n)k Fr(suc)m(h)h(that)f Fm(c)p Fk(\()p Fm(\016)t Fk(\))29 b Fp(\024)i Fm(c)1393 3075 y Fn(m)1459 3060 y Fk(\()p Fm(\016)c Fp(\000)d Fm(")p Fk(\))33 b Fr(and)h Fm(p)2025 3075 y Fn(m)2091 3060 y Fk(\()p Fm(\016)27 b Fp(\000)c Fm(")p Fk(\))30 b Fp(\024)g Fm(p)p Fk(\()p Fm(\016)d Fp(\000)c Fk(2)p Fm(")p Fk(\))p Fr(.)47 b(Note)33 b(that)h(w)m(e)h(ha)m(v)m(e)-9 3204 y Fm(s)37 3219 y Fn(n)83 3204 y Fk(\()p Fm(A)p Fk(\))28 b Fp(\025)g Fm(s)411 3219 y Fn(n)458 3204 y Fk(\()p Fm(A)569 3219 y Fn(m)636 3204 y Fk(\))k Fr(and)h(th)m(us)74 3447 y Fo(P)17 b Fk([)q Fm(s)224 3462 y Fn(n)271 3447 y Fk(\()p Fm(A)p Fk(\))28 b Fp(\024)g Fm(c)p Fk(\()p Fm(\016)t Fk(\)])f Fp(\024)i Fo(P)17 b Fk([)q Fm(s)1028 3462 y Fn(n)1075 3447 y Fk(\()p Fm(A)1186 3462 y Fn(m)1253 3447 y Fk(\))27 b Fp(\024)i Fm(c)p Fk(\()p Fm(\016)t Fk(\)])e Fp(\024)h Fo(P)17 b Fk([)r Fm(s)1899 3462 y Fn(n)1946 3447 y Fk(\()p Fm(A)2057 3462 y Fn(m)2123 3447 y Fk(\))28 b Fp(\024)g Fm(c)2336 3462 y Fn(m)2403 3447 y Fk(\()p Fm(\016)e Fp(\000)c Fm(")p Fk(\)])28 b Fp(\024)g Fm(p)2902 3462 y Fn(m)2968 3447 y Fk(\()p Fm(\016)t Fp(\000)p Fm(")p Fk(\))g Fp(\024)g Fm(p)p Fk(\()p Fm(\016)t Fp(\000)p Fk(2)p Fm(")p Fk(\))p Fm(:)-9 3654 y Fr(Letting)j Fm(")h Fr(tend)i(to)e(zero)h(yields)f(the)h (statemen)m(t.)1845 b Fc(\003)115 3848 y Fr(W)-8 b(e)33 b(arriv)m(e)f(at)h(our)f(main)f(lo)m(w)m(er)h(b)s(ound.)-9 4064 y SDict begin H.S end -9 4064 a -9 4064 a SDict begin 17.37396 H.A end -9 4064 a -9 4064 a SDict begin [/View [/XYZ H.V]/Dest (thm.19) cvn /DEST pdfmark end -9 4064 a 0 TeXcolorgray FE(Lemma)39 b(19.)p 0 TeXcolorgray 41 w Fh(L)-5 b(et)35 b Fm(\033)d Fp(2)c Fm(`)962 4079 y Fl(2)1036 4064 y Fh(b)-5 b(e)35 b(non-incr)-5 b(e)g(asing)33 b(and)h(let)h Fm(n;)17 b(k)30 b Fp(2)f Fo(N)49 b Fh(b)-5 b(e)35 b(such)g(that)g Fm(\033)3116 4079 y Fn(k)3187 4064 y Fp(6)p Fk(=)27 b(0)p Fh(.)45 b(De\034ne)1319 4271 y Fm(C)1389 4286 y Fn(k)1459 4271 y Fk(:=)28 b Fm(C)1660 4286 y Fn(k)1703 4271 y Fk(\()p Fm(\033)t Fk(\))f(=)h Fm(\033)2028 4230 y Fi(\000)p Fl(2)2024 4299 y Fn(k)2139 4177 y Fj(X)2148 4389 y Fn(j)t(>k)2299 4271 y Fm(\033)2358 4230 y Fl(2)2354 4296 y Fn(j)2414 4271 y Fm(:)-9 4557 y Fh(Then,)33 b(for)i(al)5 b(l)34 b Fm(\016)e Fp(2)c Fk(\(0)p Fm(;)17 b Fk(1\))p Fh(,)34 b(we)g(have)335 4840 y Fo(P)411 4670 y Fj(")471 4840 y Fp(R)555 4799 y Fl(\()p Fn(k)r Fl(\))555 4865 y Fn(n)653 4840 y Fk(\()p Fm(\033)t Fk(\))44 b Fp(\024)h Fm(\033)1009 4855 y Fn(k)1068 4670 y Fj( )1147 4840 y Fk(1)39 b Fp(\000)1351 4661 y Fj(s)p 1451 4661 426 4 v 1488 4773 a Fk(\(1)22 b(+)g Fm(\016)t Fk(\))p Fm(n)p 1461 4818 406 4 v 1461 4909 a Fk(\(1)f Fp(\000)i Fm(\016)t Fk(\))p Fm(C)1824 4924 y Fn(k)1893 4670 y Fj(!#)2074 4840 y Fp(\024)45 b Fk(5)17 b(exp)2427 4760 y Fj(\000)2473 4840 y Fp(\000)p Fk(\()p Fm(\016)t(=)p Fk(4\))2771 4799 y Fl(2)2843 4840 y Fk(min)e Fp(f)p Fm(n;)i(C)3244 4855 y Fn(k)3286 4840 y Fp(g)3336 4760 y Fj(\001)3399 4840 y Fm(:)p 0 TeXcolorgray -9 5128 a Fh(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(First)25 b(note)i(that,)h(in)e(the)h(setting)g (of)f(Prop)s(osition)2174 5128 y SDict begin H.S end 2174 5128 a 0 TeXcolorgray Fr(14)p 0 TeXcolorgray 2272 5063 a SDict begin H.R end 2272 5063 a 2272 5128 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.14) cvn H.B /ANN pdfmark end 2272 5128 a Fr(,)i(the)f(matrix)e Fk(\006)2870 5092 y Fi(>)2870 5157 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(n)p Fl([)p Fn(k)r Fl(])3164 5128 y Fr(and)i(the)g(v)m(ector)-9 5272 y Fk(\()p Fm(g)76 5287 y Fn(ik)142 5272 y Fk(\))180 5236 y Fn(n)180 5297 y(i)p Fl(=1)331 5272 y Fr(are)32 b(indep)s(enden)m(t.)45 b(Lemma)1432 5272 y SDict begin H.S end 1432 5272 a 0 TeXcolorgray Fr(15)p 0 TeXcolorgray 1529 5208 a SDict begin H.R end 1529 5208 a 1529 5272 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.15) cvn H.B /ANN pdfmark end 1529 5272 a 33 w Fr(and)33 b(Lemma)2100 5272 y SDict begin H.S end 2100 5272 a 0 TeXcolorgray Fr(18)p 0 TeXcolorgray 2197 5208 a SDict begin H.R end 2197 5208 a 2197 5272 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.18) cvn H.B /ANN pdfmark end 2197 5272 a 33 w Fr(yield)906 5479 y Fp(k)p Fk(\()p Fm(g)1041 5494 y Fn(ik)1107 5479 y Fk(\))1145 5438 y Fn(n)1145 5503 y(i)p Fl(=1)1263 5479 y Fp(k)1313 5494 y Fl(2)1397 5479 y Fp(\024)1519 5393 y(p)p 1602 5393 216 4 v 86 x Fk(1)22 b(+)g Fm(\016)1834 5402 y Fp(p)p 1917 5402 59 4 v 77 x Fm(n)196 b Fr(and)815 5670 y Fm(s)861 5685 y Fn(n)924 5590 y Fj(\000)970 5670 y Fk(\006)1040 5629 y Fi(>)1040 5695 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(n)p Fl([)p Fn(k)r Fl(])1307 5590 y Fj(\001)1397 5670 y Fp(\025)1519 5585 y(p)p 1602 5585 218 4 v 85 x Fk(1)22 b Fp(\000)g Fm(\016)f(\033)1891 5685 y Fn(k)1934 5579 y Fj(p)p 2034 5579 113 4 v 91 x Fm(C)2104 5685 y Fn(k)2185 5670 y Fp(\000)2301 5584 y(p)p 2384 5584 216 4 v 86 x Fk(1)h(+)g Fm(\016)e(\033)2671 5685 y Fn(k)r Fl(+1)2805 5594 y Fp(p)p 2888 5594 59 4 v 76 x Fm(n)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(16)p 0 TeXcolorgray -9 121 a Fr(with)32 b(probabilit)m(y)e(at)i(least)g Fk(1)22 b Fp(\000)h Fk(5)17 b(exp\()p Fp(\000)p Fk(\()p Fm(\016)t(=)p Fk(4\))1779 85 y Fl(2)1852 121 y Fk(min)n Fp(f)p Fm(n;)g(C)2236 136 y Fn(k)2279 121 y Fp(g)p Fk(\))p Fr(.)43 b(Note)32 b(that)h(w)m(e)g(ha) m(v)m(e)910 328 y Fm(s)956 343 y Fn(n)1019 247 y Fj(\000)1065 328 y Fk(\006)1135 343 y Fl([)p Fn(n)p Fl(])p Fn(;)p Fg(N)p Fi(nf)p Fn(k)r Fi(g)1434 247 y Fj(\001)1507 328 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17.37396 H.A end -9 2519 a -9 2519 a SDict begin [/View [/XYZ H.V]/Dest (subsection.3.3) cvn /DEST pdfmark end -9 2519 a 120 x Fr(3.3.)48 b FE(Corollaries.)i Fr(In)40 b(order)g(to)g(optimize)f(the)h(lo)m(w)m(er)g(b)s(ound)h(of)f (Theorem)3005 2639 y SDict begin H.S end 3005 2639 a 0 TeXcolorgray Fr(5)p 0 TeXcolorgray 3054 2574 a SDict begin H.R end 3054 2574 a 3054 2639 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.5) cvn H.B /ANN pdfmark end 3054 2639 a Fr(,)i(w)m(e)f(ma)m(y)f(c)m(ho)s(ose)-9 2783 y Fm(k)30 b Fp(2)e Fo(N)39 b Fr(suc)m(h)24 b(that)f(the)h(righ)m(t-hand)e (side)i(of)f(our)g(lo)m(w)m(er)g(b)s(ound)h(b)s(ecomes)f(maximal.)38 b(If)23 b(the)h(Euclidean)-9 2927 y(norm)k(of)h Fm(\033)34 b Fr(is)29 b(large,)g(w)m(e)i(simply)d(c)m(ho)s(ose)i Fm(k)h Fk(=)d(1)p Fr(.)42 b(T)-8 b(aking)29 b(in)m(to)g(accoun)m(t)h (that)g Fp(R)3024 2942 y Fn(n)3071 2927 y Fk(\()p Fm(\033)t Fk(\))g Fr(is)f(decreasing)-9 3071 y(in)i Fm(n)p Fr(,)i(w)m(e)h 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y Fk(\(1)21 b Fp(\000)i Fm(")p Fk(\))34 b Fh(for)h(al)5 b(l)35 b Fm(n)27 b Fp(\024)i Fm(n)1610 3862 y Fl(0)1684 3847 y Fh(with)35 b(pr)-5 b(ob)g(ability)34 b(at)h(le)-5 b(ast)35 b Fk(1)22 b Fp(\000)g Fk(5)p Fm(e)2968 3811 y Fi(\000)p Fn(n)3066 3820 y Fd(0)3101 3811 y Fn(=)p Fl(64)3211 3847 y Fh(.)115 4064 y Fr(This)34 b(leads)g(to)g(a)f(pro)s (of)g(of)h(Corollary)1589 4064 y SDict begin H.S end 1589 4064 a 0 TeXcolorgray Fr(6)p 0 TeXcolorgray 1638 4000 a SDict begin H.R end 1638 4000 a 1638 4064 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.6) cvn H.B /ANN pdfmark end 1638 4064 a 34 w Fr(whic)m(h)g(states)h(that)f(random)f(information)e (is)i(useful)h(if)-9 4208 y(and)e(only)g(if)g Fm(\033)f Fp(2)d Fm(`)705 4223 y Fl(2)745 4208 y Fr(.)p 0 TeXcolorgray -9 4402 a FE(Pro)s(of)37 b(of)g(Corollary)925 4402 y SDict begin H.S end 925 4402 a 0 TeXcolorgray FE(6)p 0 TeXcolorgray 981 4338 a SDict begin H.R end 981 4338 a 981 4402 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.6) cvn H.B /ANN pdfmark end 981 4402 a FE(.)p 0 TeXcolorgray 42 w Fr(W)-8 b(e)32 b(\034rst)g(consider)g(the)g(case)g(that)g Fm(\033)g Fp(2)c Fm(`)2605 4417 y Fl(2)2644 4402 y Fr(.)43 b(Since)32 b Fp(R)3052 4417 y Fn(n)3099 4402 y Fk(\()p Fm(\033)t Fk(\))c Fp(\024)g Fm(\033)3422 4417 y Fl(1)3462 4402 y Fr(,)k(Theo-)-9 4546 y(rem)186 4546 y SDict begin H.S end 186 4546 a 0 TeXcolorgray Fr(3)p 0 TeXcolorgray 234 4482 a SDict begin H.R end 234 4482 a 234 4546 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.3) cvn H.B /ANN pdfmark end 234 4546 a 33 w Fr(yields)886 4742 y Fo(E)12 b Fk([)p Fp(R)1058 4757 y Fn(n)1111 4742 y Fk(\()p Fm(\033)t Fk(\)])28 b Fp(\024)g Fk(2)p Fm(e)1500 4701 y Fi(\000)p Fn(n=)p Fl(100)1765 4742 y Fp(\001)22 b Fm(\033)1870 4757 y Fl(1)1932 4742 y Fk(+)2040 4675 y(156)p 2040 4719 147 4 v 2043 4739 a Fp(p)p 2125 4739 59 4 v 2125 4811 a Fm(n)2196 4602 y Fj(\022)2346 4647 y(X)2286 4863 y Fn(j)t Fi(\025b)p Fn(n=)p Fl(4)p Fi(c)2566 4742 y Fm(\033)2625 4701 y Fl(2)2621 4767 y Fn(j)2664 4602 y Fj(\023)2738 4624 y Fl(1)p Fn(=)p Fl(2)2848 4742 y Fm(:)-9 4999 y Fr(The)33 b(statemen)m(t)g(is)f(no)m(w)h(implied)d(b)m (y)j(the)g(fact)g(that)f Fm(\033)g Fp(2)c Fm(`)2221 5014 y Fl(2)2260 4999 y Fr(.)115 5143 y(F)-8 b(or)33 b(the)i(case)g(that)e Fm(\033)h Fp(62)d Fm(`)1108 5158 y Fl(2)1147 5143 y Fr(,)k(let)e Fk(0)d Fm(<)g(")g(<)g Fk(1)p Fr(.)47 b(F)-8 b(or)34 b Fm(m)c Fp(2)h Fo(N)49 b Fr(let)33 b Fm(\033)2537 5107 y Fl(\()p Fn(m)p Fl(\))2692 5143 y Fr(b)s(e)i(the)f(sequence)j (obtained)-9 5287 y(from)h Fm(\033)43 b Fr(b)m(y)d(replacing)f(the)g Fm(j)6 b Fr(th)40 b(elemen)m(t)f(with)g(zero)h(for)f(all)e Fm(j)46 b(>)39 b(m)p Fr(.)64 b(F)-8 b(or)39 b(an)m(y)h Fm(N)50 b Fp(\025)40 b Fm(n)p Fr(,)h(w)m(e)g(can)-9 5431 y(c)m(ho)s(ose)33 b Fm(m)28 b Fp(2)g Fo(N)47 b Fr(suc)m(h)34 b(that)1589 5539 y Fm(")1635 5503 y Fl(2)p 1558 5584 148 4 v 1558 5675 a Fk(3)p Fm(\033)1666 5641 y Fl(2)1662 5699 y(1)1773 5482 y Fn(m)1732 5512 y Fj(X)1743 5722 y Fn(j)t Fl(=2)1893 5607 y Fm(\033)1952 5565 y Fl(2)1948 5631 y Fn(j)2019 5607 y Fp(\025)28 b Fm(N)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.17) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(17)p 0 TeXcolorgray -9 121 a Fr(since)32 b Fm(\033)g Fp(62)c Fm(`)451 136 y Fl(2)491 121 y Fr(.)43 b(The)34 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Fc(\003)-9 1890 y SDict begin H.S end -9 1890 a -9 1890 a SDict begin 17.37396 H.A end -9 1890 a -9 1890 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.2) cvn /DEST pdfmark end -9 1890 a 120 x Fr(4.2.)48 b FE(Elemen)m(tary)38 b(lo)m(w)m(er)f(b)s(ound.)50 b Fr(In)33 b(this)f(section)h(w)m(e)g(pro) m(v)m(e)h(the)f(follo)m(wing)d(lo)m(w)m(er)i(b)s(ound.)-9 2226 y SDict begin H.S end -9 2226 a -9 2226 a SDict begin 17.37396 H.A end -9 2226 a -9 2226 a SDict begin [/View [/XYZ H.V]/Dest (thm.24) cvn /DEST pdfmark end -9 2226 a 0 TeXcolorgray FE(Prop)s(osition)42 b(24.)p 0 TeXcolorgray 44 w Fh(F)-7 b(or)39 b(any)g Fm(")c Fp(2)h Fk(\(0)p Fm(;)17 b Fk(1\))38 b Fh(and)h Fm(c)c(>)h Fk(0)j Fh(ther)-5 b(e)39 b(is)g(a)g(c)-5 b(onstant)38 b Fm(c)2955 2190 y Fi(0)3014 2226 y Fm(>)e Fk(0)j Fh(such)g(that)g(the)-9 2370 y(fol)5 b(lowing)36 b(holds.)53 b(If)37 b Fm(\033)865 2385 y Fn(m)965 2370 y Fp(\025)c Fm(cm)1202 2334 y Fi(\000)p Fn(\013)1345 2370 y Fh(for)38 b(some)e Fm(m)e Fp(2)f Fo(N)53 b Fh(and)37 b Fm(\013)d Fp(2)g Fk(\(0)p Fm(;)17 b Fk(1)p Fm(=)p Fk(2\))36 b Fh(then)i Fp(R)3127 2385 y Fn(n)3174 2370 y Fk(\()p Fm(\033)t Fk(\))33 b Fp(\025)h Fm(c)3495 2334 y Fi(0)3556 2370 y Fh(holds)-9 2515 y(for)g(al)5 b(l)35 b Fm(n)28 b(<)f(m)560 2478 y Fl(1)p Fi(\000)p Fl(2)p Fn(\013)770 2515 y Fh(with)35 b(pr)-5 b(ob)g(ability)34 b(at)h(le)-5 b(ast)35 b Fk(1)22 b Fp(\000)g Fm(")p Fh(.)115 2731 y Fr(T)-8 b(o)37 b(obtain)f(this)g(lo)m(w)m(er)h(b)s(ound,)h(w)m (e)g(\034rst)f(consider)g(the)g(problem)f(of)g(just)h(reco)m(v)m(ering) g(the)h(\034rst)-9 2875 y(co)s(ordinate)30 b Fm(x)524 2890 y Fl(1)596 2875 y Fr(of)h Fm(x)h Fr(in)f(the)h(unit)f(ball)e Fo(B)1527 2839 y Fn(m)1527 2900 y Fl(2)1631 2875 y Fr(of)i Fo(R)1807 2839 y Fn(m)1880 2875 y Fr(.)43 b(The)32 b(corresp)s(onding)g (radius)f(of)g(information)e(is)-9 3019 y(giv)m(en)j(b)m(y)818 3137 y Fj(f)797 3164 y Fk(rad\()p Fm(G)1053 3179 y Fn(n)1100 3164 y Fm(;)17 b Fo(B)1204 3123 y Fn(m)1204 3188 y Fl(2)1277 3164 y Fk(\))1315 3123 y Fl(2)1382 3164 y Fk(=)27 b(sup)18 b Fp(f)p Fm(x)1754 3179 y Fl(1)1804 3164 y Fk(:)50 b Fp(k)p Fm(x)p Fp(k)2036 3193 y Fl(2)2104 3164 y Fk(=)27 b(1)32 b Fr(with)h Fm(G)2588 3179 y Fn(n)2634 3164 y Fm(x)c Fk(=)e(0)p Fp(g)16 b Fm(:)-9 3380 y SDict begin H.S end -9 3380 a -9 3380 a SDict begin 17.37396 H.A end -9 3380 a -9 3380 a SDict begin [/View [/XYZ H.V]/Dest (thm.25) cvn /DEST pdfmark end -9 3380 a 0 TeXcolorgray FE(Lemma)39 b(25.)p 0 TeXcolorgray 41 w Fh(F)-7 b(or)34 b Fm(n)28 b(<)g(m)p Fh(,)35 b(we)f(have)-9 3596 y Fr(\(4\))1297 3596 y SDict begin H.S end 1297 3596 a 1297 3596 a SDict begin 17.37396 H.A end 1297 3596 a 1297 3596 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 1297 3596 a Fo(E)1375 3515 y Fj(\002)1442 3570 y(f)1422 3596 y Fk(rad\()p Fm(G)1678 3611 y Fn(n)1724 3596 y Fm(;)17 b Fo(B)1829 3555 y Fn(m)1829 3621 y Fl(2)1901 3596 y Fk(\))1939 3555 y Fl(2)1978 3515 y Fj(\003)2048 3596 y Fk(=)2161 3528 y Fm(m)23 b Fp(\000)f Fm(n)p 2161 3573 266 4 v 2251 3664 a(m)2436 3596 y(:)-9 3810 y Fh(In)34 b(p)-5 b(articular,)34 b(for)h(any)g Fm(")27 b Fp(2)h Fk(\(0)p Fm(;)17 b Fk(1\))o Fh(,)35 b(we)f(have)-9 4017 y Fr(\(5\))1375 4017 y SDict begin H.S end 1375 4017 a 1375 4017 a SDict begin 17.37396 H.A end 1375 4017 a 1375 4017 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 1375 4017 a 1395 3991 a Fj(f)1375 4017 y Fk(rad\()p Fm(G)1631 4032 y Fn(n)1678 4017 y Fm(;)17 b Fo(B)1782 3976 y Fn(m)1782 4041 y Fl(2)1854 4017 y Fk(\))1892 3976 y Fl(2)1959 4017 y Fp(\025)28 b Fk(1)22 b Fp(\000)2281 3949 y Fm(n)p 2245 3994 131 4 v 2245 4085 a("m)-9 4225 y Fh(with)34 b(pr)-5 b(ob)g(ability)35 b(at)g(le)-5 b(ast)34 b Fk(1)22 b Fp(\000)h Fm(")p Fh(.)p 0 TeXcolorgray -9 4441 a(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 41 w Fr(Let)28 b Fm(k)j Fk(=)d Fm(m)14 b Fp(\000)g Fm(n)p Fr(.)43 b(T)-8 b(o)28 b(pro)m(v)m(e)i(\()1399 4441 y SDict begin H.S end 1399 4441 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 1448 4377 a SDict begin H.R end 1448 4377 a 1448 4441 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 1448 4441 a Fr(\),)f(w)m(e)h(observ)m(e)g(that)e(w)m(e)i(w)m(an)m(t)f (to)g(compute)f(the)h(exp)s(ectation)-9 4585 y(of)j(the)h(random)e(v)-5 b(ariable)875 4766 y Fj(f)854 4792 y Fk(rad\()p Fm(G)1110 4807 y Fn(n)1157 4792 y Fm(;)17 b Fo(B)1261 4751 y Fn(m)1261 4817 y Fl(2)1334 4792 y Fk(\))1372 4751 y Fl(2)1439 4792 y Fk(=)27 b(max)1741 4712 y Fj(\010)1815 4792 y Fp(h)p Fm(x;)17 b(y)t Fp(i)27 b Fk(:)33 b Fm(x)28 b Fp(2)g Fm(E)6 b(;)34 b Fp(k)o Fm(x)p Fp(k)2602 4821 y Fl(2)2669 4792 y Fk(=)27 b(1)2821 4712 y Fj(\011)2879 4792 y Fm(;)-9 4999 y Fr(where)36 b Fm(E)41 b Fr(is)34 b(uniformly)f(distributed)h(on) h Fo(G)1646 5014 y Fn(m;k)1811 4999 y Fr(and)g Fm(y)g Fk(=)d Fm(e)2239 4963 y Fl(\(1\))2368 4999 y Fr(is)j(\034xed.)51 b(In)m(v)m(olving)35 b(an)g(orthogonal)-9 5143 y(transformation)30 b(of)i(the)h(co)s(ordinate)e(system,)j(w)m(e)f(ma)m(y)g(also)e(\034x)j (the)f(subspace)1478 5350 y Fm(E)h Fk(=)1688 5270 y Fj(\012)1735 5350 y Fm(e)1780 5309 y Fl(\(1\))1874 5350 y Fm(;)17 b(:)g(:)g(:)f(;)h(e)2138 5309 y Fl(\()p Fn(k)r Fl(\))2235 5270 y Fj(\013)-9 5557 y Fr(and)45 b(assume)g(that)g Fm(y)i Fr(is)e(uniformly)e(distributed)h(on)h(the)g(sphere.)82 b(This)45 b(do)s(es)g(not)g(c)m(hange)h(the)-9 5701 y(distribution)21 b(of)636 5675 y Fj(f)615 5701 y Fk(rad\()p Fm(G)871 5716 y Fn(n)918 5701 y Fm(;)c Fo(B)1022 5665 y Fn(m)1022 5726 y Fl(2)1095 5701 y Fk(\))1133 5665 y Fl(2)1172 5701 y Fr(.)40 b(By)24 b(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)k(inequalit)m(y) -8 b(,)24 b(the)g(maxim)m(um)d(is)i(attained)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 TeXcolorgray 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 -18 -153 a SDict begin H.S end -18 -153 a -18 -153 a SDict begin H.R end -18 -153 a -18 -153 a SDict begin [/View [/XYZ H.V]/Dest (page.22) cvn /DEST pdfmark end -18 -153 a 0 TeXcolorgray 566 -78 a Fs(RANDOM)32 b(SECTIONS)g(OF)i(ELLIPSOIDS)d(AND)h(RANDOM)g(INF)n(ORMA)-6 b(TION)496 b(22)p 0 TeXcolorgray -9 121 a Fr(for)1569 271 y Fm(x)28 b Fk(=)1835 204 y Fm(P)1898 219 y Fn(E)1958 204 y Fk(\()p Fm(y)t Fk(\))p 1766 248 389 4 v 1766 340 a Fp(k)p Fm(P)1879 355 y Fn(E)1938 340 y Fk(\()p Fm(y)t Fk(\))p Fp(k)2115 369 y Fl(2)2164 271 y Fm(;)-9 477 y Fr(where)33 b Fm(P)335 492 y Fn(E)427 477 y Fr(denotes)h(the)f (orthogonal)e(pro)5 b(jection)32 b(on)g Fm(E)6 b Fr(.)44 b(W)-8 b(e)33 b(obtain)1158 722 y Fj(f)1137 748 y Fk(rad\()p Fm(G)1393 763 y Fn(n)1440 748 y Fm(;)17 b Fo(B)1544 707 y Fn(m)1544 772 y Fl(2)1617 748 y Fk(\))1655 707 y Fl(2)1722 748 y Fk(=)27 b Fp(k)p Fm(P)1938 763 y Fn(E)1998 748 y Fk(\()p Fm(y)t Fk(\))p Fp(k)2174 699 y Fl(2)2174 777 y(2)2242 748 y Fk(=)2398 623 y Fn(k)2345 653 y Fj(X)2356 863 y Fn(j)t Fl(=1)2506 748 y Fm(y)2558 707 y Fl(2)2554 772 y Fn(j)2596 748 y Fm(:)-9 1036 y Fr(W)-8 b(e)35 b(observ)m(e)i (that)e Fo(E)13 b Fk(\()p Fm(y)876 1000 y Fl(2)872 1061 y Fn(j)920 1036 y Fk(\))33 b(=)f(1)p Fm(=m)j Fr(for)g(all)e Fm(j)39 b Fp(\024)33 b Fm(m)p Fr(,)j(since)g(these)g(terms)g(are)f (equal)g(and)g(sum)h(up)f(to)-9 1180 y(1.)54 b(This)37 b(sho)m(ws)h(\()670 1180 y SDict begin H.S end 670 1180 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 718 1116 a SDict begin H.R end 718 1116 a 718 1180 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 718 1180 a Fr(\).)55 b(Estimate)36 b(\()1294 1180 y SDict begin H.S end 1294 1180 a 0 TeXcolorgray Fr(5)p 0 TeXcolorgray 1342 1116 a SDict begin H.R end 1342 1116 a 1342 1180 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 1342 1180 a Fr(\))h(is)f(a)g(direct)g (consequence)j(of)k(\()2598 1180 y SDict begin H.S end 2598 1180 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 2647 1116 a SDict begin H.R end 2647 1116 a 2647 1180 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 2647 1180 a Fr(\))36 b(taking)g(in)m(to)f(accoun)m(t)i(that)-9 1324 y Fk(0)27 b Fp(\024)193 1298 y Fj(f)172 1324 y Fk(rad\()p Fm(G)428 1339 y Fn(n)475 1324 y Fm(;)17 b Fo(B)579 1288 y Fn(m)579 1349 y Fl(2)652 1324 y Fk(\))27 b Fp(\024)h Fk(1)p Fr(.)2795 b Fc(\003)p 0 TeXcolorgray -9 1518 a Fh(Pr)-5 b(o)g(of)34 b(of)h(Pr)-5 b(op)g(osition)885 1538 y SDict begin H.S end 885 1538 a 0 TeXcolorgray -20 x Fh(24)p 0 TeXcolorgray 984 1454 a SDict begin H.R end 984 1454 a 984 1518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (thm.24) cvn H.B /ANN pdfmark end 984 1518 a Fh(.)p 0 TeXcolorgray 42 w Fr(If)32 b(w)m(e)i(c)m(ho)s(ose)f Fm(G)1683 1533 y Fn(n)1763 1518 y Fr(satisfying)g(\()2237 1518 y SDict begin H.S end 2237 1518 a 0 TeXcolorgray Fr(5)p 0 TeXcolorgray 2285 1454 a SDict begin H.R end 2285 1454 a 2285 1518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 2285 1518 a Fr(\),)g(b)m(y)g (de\034nition)f(of)3084 1492 y Fj(f)3063 1518 y Fk(rad\()p Fm(G)3319 1533 y Fn(n)3366 1518 y Fm(;)17 b Fo(B)3470 1482 y Fn(m)3470 1543 y Fl(2)3542 1518 y Fk(\))33 b Fr(and)-9 1663 y(compactness)44 b(of)f(the)g(unit)g(sphere)i(of)d Fo(R)1585 1627 y Fn(m)1658 1663 y Fr(,)k(w)m(e)e(\034nd)g Fm(x)j Fp(2)f Fo(R)2371 1627 y Fn(m)2487 1663 y Fr(with)d Fp(k)p Fm(x)p Fp(k)2875 1678 y Fl(2)2960 1663 y Fk(=)j(1)d Fr(and)g Fm(G)3451 1678 y Fn(n)3498 1663 y Fm(x)j Fk(=)g(0)-9 1807 y Fr(satisfying)1592 1951 y Fm(x)1647 1910 y Fl(2)1647 1976 y(1)1715 1951 y Fp(\025)28 b Fk(1)22 b Fp(\000)2037 1884 y Fm(n)p 2000 1928 131 4 v 2000 2020 a("m)2141 1951 y(:)-9 2128 y Fr(Th)m(us)35 b Fm(x)294 2143 y Fl(1)368 2128 y Fr(is)e(already)h(rather)g(close)f(to)h Fk(1)g Fr(whic)m(h)g(implies)e(that)h(the)i(other)f(co)s(ordinates)f(can)h (not)g(b)s(e)-9 2272 y(to)s(o)d(big.)43 b(Indeed)34 b(w)m(e)f(\034nd) 499 2494 y Fp(k)p Fm(x)p Fp(k)654 2453 y Fl(2)654 2519 y Fn(\033)728 2494 y Fp(\024)c Fm(x)889 2453 y Fl(2)889 2519 y(1)951 2494 y Fk(+)1095 2427 y(1)p 1059 2471 123 4 v 1059 2562 a Fm(\033)1118 2533 y Fl(2)1114 2587 y Fn(m)1191 2494 y Fk(\(1)21 b Fp(\000)i Fm(x)1454 2453 y Fl(2)1454 2519 y(1)1494 2494 y Fk(\))28 b Fp(\024)g Fk(1)22 b(+)1880 2427 y(1)p 1844 2471 V 1844 2562 a Fm(\033)1903 2533 y Fl(2)1899 2587 y Fn(m)2022 2427 y Fm(n)p 1986 2471 131 4 v 1986 2562 a("m)2154 2494 y Fp(\024)28 b Fk(1)22 b(+)2583 2427 y(1)p 2438 2471 338 4 v 2438 2562 a Fm("m)2569 2533 y Fl(2)p Fn(\013)2654 2562 y Fm(\033)2713 2533 y Fl(2)2709 2587 y Fn(m)2814 2494 y Fp(\024)28 b Fk(1)22 b(+)3137 2427 y(1)p 3098 2471 128 4 v 3098 2562 a Fm("c)3186 2533 y Fl(2)3235 2494 y Fm(:)-9 2726 y Fr(Rescaling)31 b Fm(x)i Fr(yields)1380 2900 y Fk(rad\()p Fm(G)1636 2915 y Fn(n)1683 2900 y Fm(;)17 b Fp(E)1789 2859 y Fn(m)1780 2925 y(\033)1855 2900 y Fk(\))1893 2859 y Fl(2)1960 2900 y Fp(\025)2159 2833 y Fm("c)2247 2797 y Fl(2)p 2075 2878 296 4 v 2075 2969 a Fk(1)22 b(+)g Fm("c)2332 2940 y Fl(2)-9 3085 y Fr(and)32 b(\034nishes)i(the)f(pro)s(of.)2752 b Fc(\003)p 0 TeXcolorgray -9 3301 a FE(A)m(c)m(kno)m(wledgemen)m(ts.)p 0 TeXcolorgray 42 w Fr(W)-8 b(e)35 b(thank)f(sev)m(eral)h(colleagues)e (for)g(v)-5 b(aluable)33 b(remarks)h(and)g(commen)m(ts.)-9 3445 y(P)m(art)22 b(of)f(the)i(w)m(ork)f(w)m(as)h(done)g(during)e(a)h (sp)s(ecial)f(semester)i(at)e(the)i(Erwin)f(Sc)m(hr\366)s(dinger)f(In)m (ternational)-9 3590 y(Institute)27 b(for)h(Mathematics)f(and)g(Ph)m (ysics)j(\(ESI\))e(in)f(Vienna.)41 b(W)-8 b(e)28 b(thank)g(ESI)h(for)e (the)h(hospitalit)m(y)-8 b(.)-9 3734 y(A.)40 b(Hinric)m(hs)g(and)h(D.)g 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