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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">American statistician (1914–1991)</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Wassily Hoeffding</div></th></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1914-06-12</span>)</span>June 12, 1914<br /><div style="display:inline" class="birthplace"><a href="/wiki/Mukhino,_Leningrad_Oblast" title="Mukhino, Leningrad Oblast">Mustamäki</a>, <a href="/wiki/Grand_Duchy_of_Finland" title="Grand Duchy of Finland">Grand Duchy of Finland</a>, Russian Empire</div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">February 28, 1991<span style="display:none">(1991-02-28)</span> (aged 76)<br /><div style="display:inline" class="deathplace"><a href="/wiki/Chapel_Hill,_North_Carolina" title="Chapel Hill, North Carolina">Chapel Hill</a>, <a href="/wiki/North_Carolina" title="North Carolina">North Carolina</a></div></td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category"><a href="/wiki/United_States" title="United States">American</a></td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/Berlin_University" class="mw-redirect" title="Berlin University">Berlin University</a></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/Hoeffding%27s_inequality" title="Hoeffding's inequality">Hoeffding's inequality</a>, <a href="/wiki/Hoeffding%27s_lemma" title="Hoeffding's lemma">Hoeffding's lemma</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Statistician" title="Statistician">Statistician</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/University_of_North_Carolina_at_Chapel_Hill" title="University of North Carolina at Chapel Hill">University of North Carolina at Chapel Hill</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Alfred_Klose" title="Alfred Klose">Alfred Klose</a></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><a href="/wiki/Donald_Burkholder" title="Donald Burkholder">Donald Burkholder</a></li> <li><a href="/wiki/Joan_R._Rosenblatt" title="Joan R. Rosenblatt">Joan R. Rosenblatt</a></li> <li><a href="/wiki/Nicholas_Fisher_(statistician)" title="Nicholas Fisher (statistician)">Nicholas Fisher</a></li> <li><a href="/wiki/Meyer_Dwass" title="Meyer Dwass">Meyer Dwass</a></li></ul> </div></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Wassily Hoeffding</b> (June 12, 1914 – February 28, 1991) was an American <a href="/wiki/Statistician" title="Statistician">statistician</a> and <a href="/wiki/Probability_theory" title="Probability theory">probabilist</a>. Hoeffding was one of the founders of <a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">nonparametric statistics</a>, in which Hoeffding contributed the idea and basic results on <a href="/wiki/U-statistics" class="mw-redirect" title="U-statistics">U-statistics</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, <a href="/wiki/Hoeffding%27s_inequality" title="Hoeffding's inequality">Hoeffding's inequality</a> provides an <a href="/wiki/Upper_bound" class="mw-redirect" title="Upper bound">upper bound</a> on the <a href="/wiki/Probability" title="Probability">probability</a> for the sum of <a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">random variables</a> to deviate from its <a href="/wiki/Expected_value" title="Expected value">expected value</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Personal_life">Personal life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=1" title="Edit section: Personal life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hoeffding was born in Mustamäki, Finland, (Gorkovskoye, Russia since 1940), although his place of birth is registered as St. Petersburg on his birth certificate. His father was an economist and a disciple of <a href="/wiki/Peter_Berngardovich_Struve" class="mw-redirect" title="Peter Berngardovich Struve">Peter Struve</a>, the Russian social scientist and public figure. His paternal grandparents were <a href="/wiki/Danish_people" class="mw-redirect" title="Danish people">Danish</a> and his father's uncle was the Danish philosopher <a href="/wiki/Harald_H%C3%B8ffding" title="Harald Høffding">Harald Høffding</a>. His mother, née Wedensky, had studied medicine. Both grandfathers had been engineers. In 1918 the family left Tsarskoye Selo for Ukraine and, after traveling through scenes of civil war, finally left Russia for Denmark in 1920, where Wassily entered school. </p><p>In 1924 the family settled in Berlin. Hoeffding obtained his PhD in 1940 at the <a href="/wiki/Humboldt_University_of_Berlin" title="Humboldt University of Berlin">University of Berlin</a>. He migrated with his mother to the United States in 1946. His younger brother, Oleg, became a military historian in the United States. <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Hoeffding's ashes were buried in a small cemetery on land owned by George E. Nicholson, Jr.'s family in Chatham County, NC about 11 miles south of Chapel Hill, NC. </p> <div class="mw-heading mw-heading2"><h2 id="Work">Work</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=2" title="Edit section: Work"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Wassily_Hoeffding&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">December 2009</span>)</i></span></div></td></tr></tbody></table> <p>In 1948, he introduced the concept of <a href="/wiki/U-statistic" title="U-statistic">U-statistics</a>. </p><p>See the collected works of Wassily Hoeffding.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Writings">Writings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=3" title="Edit section: Writings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Masstabinvariante Korrelationstheorie</i>, 1940</li> <li><i>On the distribution of the rank correlation coefficient t when the variates are not independent</i> in <a href="/wiki/Biometrika" title="Biometrika">Biometrika</a>, 1947</li> <li><i>A class of statistics with asymptotically normal distribution</i>, 1948</li> <li><i>A nonparametric test for independence</i>, 1948</li> <li><i>The central limit theorem for dependent random variables</i> (with <a href="/wiki/Herbert_Robbins" title="Herbert Robbins">Herbert Robbins</a>), 1948</li> <li><i>"Optimum" nonparametric tests</i>, 1951</li> <li><i>A combinatorial central limit theorem</i>, 1951</li> <li><i>The large-sample power of test based on permutations of observations</i>, 1952</li> <li><i>On the distribution of the expected values of the order statistics</i>, 1953</li> <li><i>The efficiency of tests</i> (with J. R. Rosenblatt), 1955</li> <li><i>On the distribution of the number of successes in independent trials</i>, 1956</li> <li><i>Distinguishability of sets of distributions. (The case of independent and identically distributed random variables.)</i>, (with <a href="/wiki/Jacob_Wolfowitz" title="Jacob Wolfowitz">Jacob Wolfowitz</a>), 1958</li> <li><i>Lower bounds for the expected sample size and the average risk of a sequential procedure</i>, 1960</li> <li><i>Probability inequalities for sums of bounded random variables</i>, 1963</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Hoeffding%27s_bounds&action=edit&redlink=1" class="new" title="Hoeffding's bounds (page does not exist)">Hoeffding's bounds</a></li> <li><a href="/w/index.php?title=Hoeffding%27s_C1_statistic&action=edit&redlink=1" class="new" title="Hoeffding's C1 statistic (page does not exist)">Hoeffding's C1 statistic</a></li> <li><a href="/w/index.php?title=Hoeffding%27s_decomposition&action=edit&redlink=1" class="new" title="Hoeffding's decomposition (page does not exist)">Hoeffding's decomposition</a></li> <li><a href="/wiki/Hoeffding%27s_independence_test" title="Hoeffding's independence test">Hoeffding's independence test</a></li> <li><a href="/wiki/Hoeffding%27s_inequality" title="Hoeffding's inequality">Hoeffding's inequality</a></li> <li><a href="/wiki/Hoeffding%27s_lemma" title="Hoeffding's lemma">Hoeffding's lemma</a></li> <li><a href="/w/index.php?title=Hoeffding%E2%80%93Blum%E2%80%93Kiefer%E2%80%93Rosenblatt_process&action=edit&redlink=1" class="new" title="Hoeffding–Blum–Kiefer–Rosenblatt process (page does not exist)">Hoeffding–Blum–Kiefer–Rosenblatt process</a></li> <li><a href="/w/index.php?title=Terry%E2%80%93Hoeffding_test&action=edit&redlink=1" class="new" title="Terry–Hoeffding test (page does not exist)">Terry–Hoeffding test</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Wassily Hoeffding (1948) "A class of statistics with asymptotically normal distributions". <a href="/wiki/Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a>, 19, 293–325. (Partially reprinted in: Kotz, S., Johnson, N.L. (1992) <i>Breakthroughs in Statistics</i>, Vol I, pp 308–334. Springer-Verlag. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94037-5" title="Special:BookSources/0-387-94037-5">0-387-94037-5</a>)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Pranab_K._Sen" title="Pranab K. Sen">Sen, P.K</a> (1992) "Introduction to Hoeffding (1948) A Class of Statistics with Asymptotically Normal Distribution". In: Kotz, S., Johnson, N.L. <i>Breakthroughs in Statistics</i>, Vol I, pp 299–307. Springer-Verlag. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94037-5" title="Special:BookSources/0-387-94037-5">0-387-94037-5</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Wassily Hoeffding (1963) Probability inequalities for sums of bounded random variables, <i><a href="/wiki/Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i>, 58 (301), 13–30. (<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2282952">JSTOR</a>)</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFishervan_Zwet2005" class="citation book cs1">Fisher, Nickolas J; van Zwet, Willem R (2005). <a rel="nofollow" class="external text" href="http://www.nap.edu/read/11429/chapter/12"><i>Biographic Memoirs Volume 86</i></a>. The National Academies Press, Washington D.C. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17226%2F11429">10.17226/11429</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-309-09304-0" title="Special:BookSources/978-0-309-09304-0"><bdi>978-0-309-09304-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">July 28,</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Biographic+Memoirs+Volume+86&rft.pub=The+National+Academies+Press%2C+Washington+D.C.&rft.date=2005&rft_id=info%3Adoi%2F10.17226%2F11429&rft.isbn=978-0-309-09304-0&rft.aulast=Fisher&rft.aufirst=Nickolas+J&rft.au=van+Zwet%2C+Willem+R&rft_id=http%3A%2F%2Fwww.nap.edu%2Fread%2F11429%2Fchapter%2F12&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWassily+Hoeffding" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><i>The Collected Works of Wassily Hoeffding</i> (1994), N. I. Fisher and P. K. Sen, eds., Springer-Verlag, New York.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wassily_Hoeffding&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=20681">Wassily Hoeffding</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output 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