1istCryx00660069Cfx00660069PrCb6bilityx006600696n9x00660069St6tisti8sx00660069x006600695Cl ID: 123253
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JCHrn.l�Jle8trCniDHe�9M1istCire�9es�PrCb6bilitJs�et�9e�l6�St6tistiDHe/�Ele8trCni8�JCHrn6l�fCr� 1istCry�Cf�PrCb6bility�6n9�St6tisti8s�.�5Cl.,´�n°1.�JHin/JHne�2009 EbS.i9SePflnJeP,8nSl8Snf5,JlC8e6J9SnyntroductiontotheJuneJEE9issueofthe�P92e865J2´6f850P.68�J9e68l6..86101JPJel05,/e0eJ9eJ292JDlSeC,�120�1JeP.9See./�´�9Overthepastsixtyyears,martingaleshavebecomecentralinthemath9ematicsofrandomness.Theyappearinthegeneraltheoryofstochasticprocesses,inthealgorithmictheoryofrandomness,andinsomebranchesofmathematicalstatistics.Yetlittlehasbeenwrittenaboutthehistoryofthisevolution.Thisspecialissueexploressomeoftheterritorythatahistoryofmartingaleswouldneedtotraverse.Thehistorianofmartingalesfacesanimmensetask.Wecan�ndtracesofmartingalethinkingattheverybeginningofprobabilitytheory,becausethistheorybeganasastudyofgambling,andtheevolutionofagambler0sholdingsasaresultoffollowingaparticularstrategycanalwaysbeunder9stoodasamartingale.Morerecently,inthesecondhalfofthetwentiethcentury,martingalesbecameimportantinthetheoryofstochasticprocessesattheverysametimethatstochasticprocesseswerebecomingincreasinglyimportantinstatisticsandinscience.Moreover,ahistoryofmartingales,likeahistoryofanyotherbranchofmathematics,mustgofarbeyondanaccountofmathematicalideasandtechniques.ytmustexplorethecontextinwhichtheevolutionofideastookplaceethebroaderintellectualmilieuxoftheactors,thenetworksthatal9readyexistedorwerecreatedbytheresearch,eventhesocialandpolitical 9SP9156Pb�11.5P6.�05.SE128bJ1569SP9156Pbe�9´,´/´ JCHrn.l�Jle8trCniDHe�9M1istCire�9es�PrCb6bilitJs�et�9e�l6�St6tistiDHe/�Ele8trCni8�JCHrn6l�fCr� 1istCry�Cf�PrCb6bility�6n9�St6tisti8s�.�5Cl.,´�n°1.�JHin/JHne�2009 conditionsthatfavoredorhamperedthecirculationandadoptionofcertainSowedonotpretendtotreatthehistoryofmartingalesexhaustively.Wepresentonlyastrollthroughthishistory,inpartaguidedtour,inpartarandomwalk.We�rstpresenthistoricalstudiesontheperiodfrom19JEto195E,whenmartingalesemergedasadistinctmathematicalconcept.Thenweo�erinsightsontheperiodfrom195Eintothe198Es,whentheconceptshoweditsvalueinstochasticprocesses,mathematicalstatistics,andalgo9rithmicrandomness.Weo�erelevenarticles.Mostwerewrittenforthisoccasion.lfewarerevisionsofarticlesimportantforourtopic,translatedfromthesrenchforthe�rsttime.Theseelevenarticlesvarygreatlyintoneandpurpose,fortheytreatdi�erentaspectsofthehistoryofmartingalesandhaveauthorswithavarietyofperspectives.lswecomeclosertothepresent,andmoreconcernedwithideasthathavenotyetbeenfullydigestedbyallwhomightusethem,historyissometimesmixedwithexposition.nuteachofthecontributionsopenstoourviewanotherimportantlandscapewithinthevastterritorytraversedbythehistoryofmartingales.The�rstarticle,byRogerMlNSUY,takesusonacolorfuljourneyintothepastoftheword.lsMlNSUYshowsus,theword0sultimateetymologymayremainuncertain,butwecan�ndenjoyment,andsurprises,inthedi�erentmeaningsithastakenoverthecourseoftime.ynthesecondarticle,nernardnRU,MariesrancenRU,andtherecentlydeceased°aiLaiCHUNtrecountrmilenorel0sencounterwithmartingalesandplaceitinthecontextofthelife0sworkofthispowerfulmathematician,wholaunchedmodernprobabilitytheoryinsomerespects,andwhosevisionofitspracticalimportancedrovemuchofitsdevelopmentbetweenthetwoworldwars.norel�rstencounteredamathematicalde�nitionofmartingaleswhenhewasnearlybE,inthethesisworkofhisstudentandresearchassistantJeanVille,andhethenusedtheideatorevisittheSt.Petersburgparadoxandtocontinueadebateaboutthevalueofprobabilitytheorythathehadbegunnearlyfortyyearsearlierwiththebiologists´elixLeDantec.ynanotherarticle,nernardnRUandSalahryDintroduceuston.rgeJessenandPaulL´evy,whoseworkinthe19PEsanticipatedwhatwenowknowasJosephDoob0smartingaleconvergencetheorem.L´evyissometimesconsideredaninventorofmartingalesbecauseofhisworkondependentran9domvariables.Hiszero9onelaw–L´evy0slemma,asnRUandryDcallit–containsthecentralideaofmartingaleconvergenceandisalsocloselyrelatedtoJessen0sconstructionofin�nitedimensionalintegralsasthelimitof�nitedimensionalintegrals.nRUandryDbasetheirstudyonacorrespondencebetweenthetwomathematicians,initiatedbyL´evywhenhediscoveredthekinshipofhisworkwithJessen0s.TheirstudyisparticularlyimportantasanintroductiontoJessen0searlywork,toolittleknownbecauseitwasinDanish.TheappendicestotheirarticleincludepassagesfromJessen0sthe9ses,translatedfromtheDanish,andJessen0slatercorrespondencewithDoob JCHrn.l�Jle8trCniDHe�9M1istCire�9es�PrCb6bilitJs�et�9e�l6�St6tistiDHe/�Ele8trCni8�JCHrn6l�fCr� 1istCry�Cf�PrCb6bility�6n9�St6tisti8s�.�5Cl.,´�n°1.�JHin/JHne�2009 conditionsthatfavoredorhamperedthecirculationandadoptionofcertainSowedonotpretendtotreatthehistoryofmartingalesexhaustively.Wepresentonlyastrollthroughthishistory,inpartaguidedtour,inpartarandomwalk.We�rstpresenthistoricalstudiesontheperiodfrom19JEto195E,whenmartingalesemergedasadistinctmathematicalconcept.Thenweo�erinsightsontheperiodfrom195Eintothe198Es,whentheconceptshoweditsvalueinstochasticprocesses,mathematicalstatistics,andalgo9rithmicrandomness.Weo�erelevenarticles.Mostwerewrittenforthisoccasion.lfewarerevisionsofarticlesimportantforourtopic,translatedfromthesrenchforthe�rsttime.Theseelevenarticlesvarygreatlyintoneandpurpose,fortheytreatdi�erentaspectsofthehistoryofmartingalesandhaveauthorswithavarietyofperspectives.lswecomeclosertothepresent,andmoreconcernedwithideasthathavenotyetbeenfullydigestedbyallwhomightusethem,historyissometimesmixedwithexposition.nuteachofthecontributionsopenstoourviewanotherimportantlandscapewithinthevastterritorytraversedbythehistoryofmartingales.The�rstarticle,byRogerMlNSUY,takesusonacolorfuljourneyintothepastoftheword.lsMlNSUYshowsus,theword0sultimateetymologymayremainuncertain,butwecan�ndenjoyment,andsurprises,inthedi�erentmeaningsithastakenoverthecourseoftime.ynthesecondarticle,nernardnRU,MariesrancenRU,andtherecentlydeceased°aiLaiCHUNtrecountrmilenorel0sencounterwithmartingalesandplaceitinthecontextofthelife0sworkofthispowerfulmathematician,wholaunchedmodernprobabilitytheoryinsomerespects,andwhosevisionofitspracticalimportancedrovemuchofitsdevelopmentbetweenthetwoworldwars.norel�rstencounteredamathematicalde�nitionofmartingaleswhenhewasnearlybE,inthethesisworkofhisstudentandresearchassistantJeanVille,andhethenusedtheideatorevisittheSt.Petersburgparadoxandtocontinueadebateaboutthevalueofprobabilitytheorythathehadbegunnearlyfortyyearsearlierwiththebiologists´elixLeDantec.ynanotherarticle,nernardnRUandSalahryDintroduceuston.rgeJessenandPaulL´evy,whoseworkinthe19PEsanticipatedwhatwenowknowasJosephDoob0smartingaleconvergencetheorem.L´evyissometimesconsideredaninventorofmartingalesbecauseofhisworkondependentran9domvariables.Hiszero9onelaw–L´evy0slemma,asnRUandryDcallit–containsthecentralideaofmartingaleconvergenceandisalsocloselyrelatedtoJessen0sconstructionofin�nitedimensionalintegralsasthelimitof�nitedimensionalintegrals.nRUandryDbasetheirstudyonacorrespondencebetweenthetwomathematicians,initiatedbyL´evywhenhediscoveredthekinshipofhisworkwithJessen0s.TheirstudyisparticularlyimportantasanintroductiontoJessen0searlywork,toolittleknownbecauseitwasinDanish.TheappendicestotheirarticleincludepassagesfromJessen0sthe9ses,translatedfromtheDanish,andJessen0slatercorrespondencewithDoob andwithJeanDieudonn´e.ynourfourtharticle,LaurentMlZLyl°surveysL´evy0sworkrelatedtomartingalesinthe19PEsandstudiesL´evy0srelationshipwithhismuchyoungercolleagueVille,whoinventedmartingalesinadi�erentwayinthesameperiod.ytwasVille,notL´evy,whoinspiredDoob0sworkandthuslaunchedmartingalesinprobabilitytheory,butL´evyneverunderstoodVille0spurposesandneverratedhistalenthighly.Jeanlndr´eVilleisacentral�gureinthehistoryofmartingales.Hisdoc9toralthesis,wheretheconceptofamartingalewas�rstexplicitlyformulated,ismentionedinmanyofthearticlesinthisissue,butboththedetailsofhismathematicalaccomplishmentandthestoryofhislifearerelativelylittleknown.Howdidhecomeupontheconceptofamartingale,andwhydidheabandonmathematicalresearchonmartingalesitlennSHlsrR,whoiswritingapersonalandscienti�cbiographyofVille,o�ershereanaccountofVille0searlylifeandwork,fromhisbirthinMarseillein191Etohisdefenseofhisthesisin19P9.Oursixtharticle,byLaurentnyrNVrNU,tlennSHlsrR,andllexan9derSHrN,placesmartingalesinthecontextwhereVille�rstdiscoveredthemethestudyofrandomness.VillewasinspiredbytheworkofRichardvonMisesandlbrahamWald,whowantedtobaseprobabilitytheoryonade�nitionofrandomsequences,or26PP92eJi99.VonMisesandWald0scollec9tiveswerecharacterizedbylimitingfrequenciesthatdonotchangewhenthesequenceisreplacedbyasubsequencefVille0scollectiveswerecharacterizedbytheirresistancetogamblingstrategiesthattrytomultiplythecapitalriskedbyanin�nitefactor.nyrNVrNU,SHlsrR,andSHrNreviewboththeworkofWaldandVilleinthe19PEsandtherevivalofthestudyofran9domnessafterthediscovery,bylndrei°olmogorovandothersinthe196Es,oftheconceptofalgorithmiccomplexityanditsabilitytocharacterizeran9domness.Heremartingalesagainfoundtheirplace,especiallyintheworkofClaus9PeterSchnorrandLeonidLevin.nernardLOC°rRfocusesonaspeci�ceventeJosephLeoDoob0spresen9tationofhisworkonmartingalesatLyonin19S8,ataconferenceorganizedbyMauricesr´echet.HereDoobdemonstratedthepowerofhisresultsonmartingaleconvergencebyshowinghoweasilytheydealwiththestronglawoflargenumbersandcertainproblemsofstatisticalestimation.LOC°rRre9viewsthetextofDoob0spresentationandplacesitinthecontextofpreviouscontributionsbyL´evy,Ville,andDoobhimself.Thefourremainingarticlesarewrittenbymathematicianswhohavepi9oneeredtheuseofmartingalesinvariousarenas.Twoofthesearticlesarewrittenbyprobabilists,andtwoarewrittenbystatisticians.sirst,wepresentanarticlebythelatePaul9lndr´eMrYrR119PS–JEEP2.HereMrYrRreviewsthehistoryofthetheoryofstochasticprocessesfrom195Eintothe199Es.llthoughheemphasizestheaspectsofthisdevelop9mentthatinterestedhimpersonallyasaworkingmathematician,hesurveysavastterritory,andthissurveyamplydemonstratestheimportanceofmar9 JCHrn.l�Jle8trCniDHe�9M1istCire�9es�PrCb6bilitJs�et�9e�l6�St6tistiDHe/�Ele8trCni8�JCHrn6l�fCr� 1istCry�Cf�PrCb6bility�6n9�St6tisti8s�.�5Cl.,´�n°1.�JHin/JHne�2009 tingales.The�rststepoftheirdevelopment,afterDoobhadexhibitedtheirpower,wasthegeneralization,inthe195Es,ofytˆo0sstochasticintegrationfornrownianmotion.Themartingaleproperty,generalizedinseveraldirections,especiallyintheconceptsoflocalmartingaleandsemimartingale,emergedasthefundamentalpropertyrequiredforstochasticintegrationandstochasticdi�erentialequations.Thesecondstep,inwhichMrYrRandhisseminaratStrasbourgwerecentral,wastheE95980Pe1968l6.b86299999thatemergedinthe196Es.ytdemonstratedboththegeneralityofmanyideasthathad�rstbeendevelopedforMarkovprocessesandtheroleofmartingalesinverygeneralrepresentationsofstochasticprocesses.lnotherdistinguishedprobabilist,ShinzoWlTlNlnr,providessomedetailsabouthowJapanesemathematiciansbecameinvolvedinresearchinmartngales.Hebeginsofcoursewith°iyosiytˆo0searlywork,butheempha9sizesworkbythemanyJapaneseresearcherswhotookinspirationfromytˆo0scollaborationwithHenryMc°eaninthe195Esandfromtheirlecturesat°yotoafterytˆo0sreturnfromtheUnitedStatesin1956.WhileemphasizingtheJapanesecontribution,hehelpsusunderstandthedi�usionofmartingaletechniquesacrossmathematicsaswellasacrosscontinents.ynthe�rstofourarticlesbystatisticians,TzeLeungLlyreviewstheroleofmartingalesinsequentialanalysisandtimeseries,beginningwithWald0s19S5articleonthesequentialprobabilityratiotest.LlyworkedonmartingalesandsequentialanalysisatColumbiaUniversity,withYuanShihChow,HerbertRobbins,andDavidSiegmund.Heseestheemergenceofmartingalesinsequentialanalysisasaprocessthatstretchedfrom19S5to19b5,withtheirroleintimeseriesanalysisemerginglater,from19b5to1985.Thesecondarticleisbyateamofstatisticians,OddllLrN,Per°raghlNDrRSrN,Niels°ryDyNt,RichardtyLLand/rnulfnORtlN,whoseworkontheapplicationsofmartingalestosurvivalanalysisbeganwithlalen0sdoctoralthesisatnerkeleyin19b5.Theyseethesubsequent�fteenyearsasaperiodduringwhichthegeneraltheoryofprocessesdevelopedbytheStrasbourgschoolallowedtheclari�cationandimplementationofintu9itiveideasbroughtintosurvivalanalysisbytheMantel9Haenszeltest119592andDavidCox0sconceptofpartiallikelihood119b52.Tocompletethepackage,wepresentanumberofsigni�cantdocuments.Someareoriginaltexts,somedocumentsfromarchives,publicorpersonal.lgain,somewerepreviouslypublished,butnotinrnglish.sirstcomesabiographyofn.rgeJessenbyChristiannerg,basedontheobituarynergpublishedinDanishrightafterJessen0sdeathin199P.Thenashortmemoirby°laus°rickeberg,animportantcontributortothetheoryofmartingalesafterWorldWaryy.ThensixdocumentsconcerningJeanVille.Theyincludeapartlyauto9biographicalarticlebyVilleandanobituaryofVillebyhisoldfriendandcolleaguefromstudentdays,nernardd0Oregval.Othersaretranslationsfromarchivaldocuments.WecallspecialattentiontothepresentationofPierreCr´epel0sinterviewandcorrespondencewithVillein198S–85,auniquesource JCHrn.l�Jle8trCniDHe�9M1istCire�9es�PrCb6bilitJs�et�9e�l6�St6tistiDHe/�Ele8trCni8�JCHrn6l�fCr� 1istCry�Cf�PrCb6bility�6n9�St6tisti8s�.�5Cl.,´�n°1.�JHin/JHne�2009 tingales.The�rststepoftheirdevelopment,afterDoobhadexhibitedtheirpower,wasthegeneralization,inthe195Es,ofytˆo0sstochasticintegrationfornrownianmotion.Themartingaleproperty,generalizedinseveraldirections,especiallyintheconceptsoflocalmartingaleandsemimartingale,emergedasthefundamentalpropertyrequiredforstochasticintegrationandstochasticdi�erentialequations.Thesecondstep,inwhichMrYrRandhisseminaratStrasbourgwerecentral,wastheE95980Pe1968l6.b86299999thatemergedinthe196Es.ytdemonstratedboththegeneralityofmanyideasthathad�rstbeendevelopedforMarkovprocessesandtheroleofmartingalesinverygeneralrepresentationsofstochasticprocesses.lnotherdistinguishedprobabilist,ShinzoWlTlNlnr,providessomedetailsabouthowJapanesemathematiciansbecameinvolvedinresearchinmartngales.Hebeginsofcoursewith°iyosiytˆo0searlywork,butheempha9sizesworkbythemanyJapaneseresearcherswhotookinspirationfromytˆo0scollaborationwithHenryMc°eaninthe195Esandfromtheirlecturesat°yotoafterytˆo0sreturnfromtheUnitedStatesin1956.WhileemphasizingtheJapanesecontribution,hehelpsusunderstandthedi�usionofmartingaletechniquesacrossmathematicsaswellasacrosscontinents.ynthe�rstofourarticlesbystatisticians,TzeLeungLlyreviewstheroleofmartingalesinsequentialanalysisandtimeseries,beginningwithWald0s19S5articleonthesequentialprobabilityratiotest.LlyworkedonmartingalesandsequentialanalysisatColumbiaUniversity,withYuanShihChow,HerbertRobbins,andDavidSiegmund.Heseestheemergenceofmartingalesinsequentialanalysisasaprocessthatstretchedfrom19S5to19b5,withtheirroleintimeseriesanalysisemerginglater,from19b5to1985.Thesecondarticleisbyateamofstatisticians,OddllLrN,Per°raghlNDrRSrN,Niels°ryDyNt,RichardtyLLand/rnulfnORtlN,whoseworkontheapplicationsofmartingalestosurvivalanalysisbeganwithlalen0sdoctoralthesisatnerkeleyin19b5.Theyseethesubsequent�fteenyearsasaperiodduringwhichthegeneraltheoryofprocessesdevelopedbytheStrasbourgschoolallowedtheclari�cationandimplementationofintu9itiveideasbroughtintosurvivalanalysisbytheMantel9Haenszeltest119592andDavidCox0sconceptofpartiallikelihood119b52.Tocompletethepackage,wepresentanumberofsigni�cantdocuments.Someareoriginaltexts,somedocumentsfromarchives,publicorpersonal.lgain,somewerepreviouslypublished,butnotinrnglish.sirstcomesabiographyofn.rgeJessenbyChristiannerg,basedontheobituarynergpublishedinDanishrightafterJessen0sdeathin199P.Thenashortmemoirby°laus°rickeberg,animportantcontributortothetheoryofmartingalesafterWorldWaryy.ThensixdocumentsconcerningJeanVille.Theyincludeapartlyauto9biographicalarticlebyVilleandanobituaryofVillebyhisoldfriendandcolleaguefromstudentdays,nernardd0Oregval.Othersaretranslationsfromarchivaldocuments.WecallspecialattentiontothepresentationofPierreCr´epel0sinterviewandcorrespondencewithVillein198S–85,auniquesource ofinformationaboutthisoftenelusiveindividual.Weincludetwoletters,onefromJosephDoobtoPierreCr´epel,andonefromPaulL´evytoTakeyukiHida.Weconcludewithaverybrief�lm,generouslyprovidedbyHida.MadebyHidainMarch1968,itprovidesaglimpseofPaulandSuzanneL´evyonthebalconyoftheirapartmentinParis.Wethankthetwentyauthors,halfdozentranslators,andmanyrefereeswhomadethisissuepossible.