Probability Theory PDF
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Universitext Alexandr A. Borovkov Probability Theory Universitext Universitext SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA VincenzoCapasso UniversitàdegliStudidiMilano,Milan,Italy CarlesCasacuberta UniversitatdeBarcelona,Barcelona,Spain AngusMacIntyre QueenMary,UniversityofLondon,London,UK KennethRibet UniversityofCalifornia,Berkeley,Berkeley,CA,USA ClaudeSabbah CNRS,ÉcolePolytechnique,Palaiseau,France EndreSüli UniversityofOxford,Oxford,UK WojborA.Woyczynski CaseWesternReserveUniversity,Cleveland,OH,USA Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal, even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolutionofteachingcurricula,intoverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Forfurthervolumes: www.springer.com/series/223 Alexandr A. Borovkov Probability Theory Edited by K.A. Borovkov Translated by O.B. Borovkova and P.S. Ruzankin AlexandrA.Borovkov SobolevInstituteofMathematicsand NovosibirskStateUniversity Novosibirsk,Russia Translationfromthe5thedn.oftheRussianlanguageedition: ‘TeoriyaVeroyatnostei’byAlexandrA.Borovkov ©KnizhnyidomLibrokom2009 AllRightsReserved. 1stand2ndedn.©Nauka1976and1986 3rdedn.©EditorialURSSandSobolevInstituteofMathematics1999 4thedn.©EditorialURSS2003 ISSN0172-5939 ISSN2191-6675(electronic) Universitext ISBN978-1-4471-5200-2 ISBN978-1-4471-5201-9(eBook) DOI10.1007/978-1-4471-5201-9 SpringerLondonHeidelbergNewYorkDordrecht LibraryofCongressControlNumber:2013941877 MathematicsSubjectClassification: 60-XX,60-01 ©Springer-VerlagLondon2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Foreword Thepresenteditionofthebookdifferssubstantiallyfromthepreviousone.Overthe periodoftimesincethepublicationofthepreviouseditiontheauthorhasaccumu- latedquitealotofideasconcerningpossibleimprovementstosomechaptersofthe book.Inaddition,somenewopportunitieswerefoundforanaccessibleexposition of new topics that had not appeared in textbooks before but which are of certain interest for applications and reflect current trends in the development of modern probability theory. All this led to the need for one more revision of the book. As a result, many methodological changes were made and a lot of new material was added, which makes the book more logically coherent and complete. We will list hereonlythemainchangesintheorderoftheirappearanceinthetext. •Section4.4“ExpectationsofSumsofaRandomNumberofRandomVariables” wassignificantlyrevised.NewsufficientconditionsforWald’sidentitywereadded. Anexampleisgivenshowingthat,whensummandsarenon-identicallydistributed, Wald’s identity can fail to hold even in the case when its right-hand side is well- defined.Lateron,Theorem11.3.2showsthat,foridenticallydistributedsummands, Wald’sidentityisalwaysvalidwheneveritsright-handsideiswell-defined. • In Sect. 6.1 a criterion of uniform integrability of random variables is con- structed,whichsimplifiestheuseofthisnotion.Forexample,thecriteriondirectly impliesuniformintegrabilityofweightedsumsofuniformlyintegrablerandomvari- ables. •Section7.2,whichisdevotedtoinversionformulas,wassubstantiallyexpanded andnowincludesassertionsusefulforprovingintegro-localtheoremsinSect.8.7. •InChap.8,integro-locallimittheoremsforsumsofidenticallydistributedran- domvariableswereadded(Sects.8.7and8.8).Thesetheorems,beingsubstantially more precise assertions than the integral limit theorems, do not require additional conditionsandplayanimportantroleininvestigatinglargedeviationprobabilities inChap.9. v vi Foreword •Anewchapterwaswrittenonprobabilitiesoflargedeviationsofsumsofran- dom variables (Chap. 9). The chapter provides a systematic and rather complete expositionofthelargedeviationtheorybothinthecasewheretheCramércondition (rapiddecayofdistributionsatinfinity)issatisfiedandwhereitisnot.Bothintegral andintegro-localtheoremsareobtained.Thelargedeviationprincipleisestablished. •Assertionsconcerningthecaseofnon-identicallydistributedrandomvariables were added in Chap. 10 on “Renewal Processes”. Among them are renewal theo- remsaswellasthelawoflargenumbersandthecentrallimittheoremforrenewal processes. A new section was written to present the theory of generalised renewal processes. • An extension of the Kolmogorov strong law of large numbers to the case of non-identically distributed random variables having the first moment only was addedtoChap.11.Anewsubsectiononthe“Stronglawoflargenumbersforgen- eralisedrenewalprocesses”waswritten. • Chapter 12 on “Random walks and factorisation identities” was substantially revised.Anumberofnewsectionswereadded:onfindingfactorisationcomponents inexplicitform,ontheasymptoticpropertiesofthedistributionofthesupremaof cumulated sums and generalised renewal processes, and on the distribution of the firstpassagetime. •InChap.13,devotedtoMarkovchains,asectionon“Thelawoflargenumbers andcentrallimittheoremforsumsofrandomvariablesdefinedonaMarkovchain” wasadded. •Threenewappendices(6,7and8)werewritten.Theypresentimportantaux- iliary material on the following topics: “The basic properties of regularly varying functionsandsubexponentialdistributions”,“Proofsoftheoremsonconvergenceto stablelaws”,and“Upperandlowerboundsforthedistributionsofsumsandmaxima ofsumsofindependentrandomvariables”. Ashasalreadybeennoted,thesearejustthemostsignificantchanges;thereare alsomanyothers.Alotoftyposandotherinaccuracieswerefixed.Theprocessof creatingnewtyposandmisprintsinthecourseofone’sworkonabookisrandom and can be well described mathematically by the Poisson process (for the defini- tionofPoissonprocesses,seeChaps10and19).Animportantcharacteristicofthe qualityofabookistheintensityofthisprocess.Unfortunately,Iamafraidthatin thetwopreviouseditions(1999and2003)thisintensityperhapsexceededacertain acceptablelevel.Notrenouncinghisownresponsibility,theauthorstilladmitsthat thismaybedue,tosomeextent,tothefactthatthepublicationoftheseeditionstook placeatthetimeofacertaindeclineofthepublishingindustryinRussiarelatedto thegeneralstateoftheeconomyatthattime(inthe1972,1976and1986editions thereweremuchfewersuchdefects). Foreword vii Beforestartingtoworkonthenewedition,Iaskedmycolleaguesfromourlab- oratory at the Sobolev Institute of Mathematics and from the Chair of Probability TheoryandMathematicalStatisticsatNovosibirskStateUniversitytopreparelists of any typos and other inaccuracies they had spotted in the book, as well as sug- gested improvements of exposition. I am very grateful to everyone who provided me with such information. I would like to express special thanks to I.S. Borisov, V.I.Lotov,A.A.Mogul’skyandS.G.Foss,whoalsoofferedanumberofmethod- ologicalimprovements. IamalsodeeplygratefultoT.V.Belyaevaforherinvaluableassistanceintype- settingthebookwithitsnumerouschanges.Withoutthathelp,theworkonthenew editionwouldhavebeenmuchmoredifficult. A.A.Borovkov Foreword to the Third and Fourth Editions This book has been written on the basis of the Russian version (1986) published by “Nauka” Publishers in Moscow. A number of sections have been substantially revised and several new chapters have been introduced. The author has striven to provideacompleteandlogicalexpositionandsimplerandmoreillustrativeproofs. The1986textwasprecededbytwoearliereditions(1972and1976).Thefirstone appeared as an extended version of lecture notes of the course the author taught attheDepartmentofMechanicsandMathematicsofNovosibirskStateUniversity. Each new edition responded to comments by the readers and was completed with newsectionswhichmadetheexpositionmoreunifiedandcomplete. Thereadersareassumedtobefamiliarwithatraditionalcalculuscourse.They would also benefit from knowing elements of measure theory and, in particular, thenotionofintegralwithrespecttoameasureonanarbitraryspaceanditsbasic properties. However, provided they are prepared to use a less general version of someoftheassertions,thislackofadditionalknowledgewillnothinderthereader fromsuccessfullymasteringthematerial.Itisalsopossibleforthereadertoavoid suchcomplicationscompletelybyreadingtherespectiveAppendices(locatedatthe endofthebook)whichcontainallthenecessaryresults. Thefirsttenchaptersofthebookaredevotedtothebasicsofprobabilitytheory (includingthemainlimittheoremsforcumulativesumsofrandomvariables),andit isbesttoreadtheminsuccession.Theremainingchaptersdealwithmorespecific parts of the theory of probability and could be divided into two blocks: random processesindiscretetime(orrandomsequences,Chaps.12and14–16)andrandom processesincontinuoustime(Chaps.17–21). Therearealsochapterswhichremainoutsidethemainstreamofthetextasindi- catedabove.TheseincludeChap.11“FactorisationIdentities”.Thechapternotonly containsaseriesofveryusefulprobabilisticresults,butalsodisplaysinterestingre- lationships between problems on random walks in the presence of boundaries and boundaryproblemsofcomplexanalysis.Chapter13“InformationandEntropy”and Chap.19“FunctionalLimitTheorems”alsodeviatefromthemainstream.Thefor- merdealswithproblemscloselyrelatedtoprobabilitytheorybutveryrarelytreated in texts on the discipline. The latter presents limit theorems for the convergence ix x ForewordtotheThirdandFourthEditions ofprocessesgeneratedbycumulativesumsofrandomvariablestotheWienerand Poissonprocesses;asaconsequence,thelawoftheiteratedlogarithmisestablished inthatchapter. The book has incorporated a number of methodological improvements. Some partsofitaredevotedtosubjectstobecoveredinatextbookforthefirsttime(for example, Chap. 16 on stochastic recursive sequences playing an important role in applications). The book can serve as a basis for third year courses for students with a rea- sonable mathematicalbackground,and also for postgraduates. A one-semester (or two-trimester)courseonprobabilitytheorymightconsist(therecouldbemanyvari- ants)ofthefollowingparts:Chaps.1–2,Sects.3.1–3.4,4.1–4.6(partially),5.2and 5.4(partially),6.1–6.3(partially),7.1,7.2,7.4–7.6,8.1–8.2and8.4(partially),10.1, 10.3,andthemainresultsofChap.12. For a more detailed exposition of some aspects of Probability Theory and the TheoryofRandomProcesses,seeforexample[2,10,12–14,26,31]. While working on the different versions of the book, I received advice and help from many of my colleagues and friends. I am grateful to Yu.V. Prokhorov, V.V. Petrov and B.A. Rogozin for their numerous useful comments which helped toimprovethefirstvariantofthebook.IamdeeplyindebtedtoA.N.Kolmogorov whoseremarksandvaluablerecommendations,especiallyofmethodologicalchar- acter, contributed to improvements in the second version of the book. In regard to thesecondandthirdversions,IamagainthankfultoV.VPetrovwhogavemehis comments,andtoP.Franken,withwhomIhadalotofusefuldiscussionswhilethe bookwastranslatedintoGerman. InconclusionIwanttoexpressmysinceregratitudetoV.V.Yurinskii,A.I.Sakha- nenko,K.A.Borovkov,andothercolleaguesofminewhoalsogavemetheircom- mentsonthemanuscript.Iwouldalsoliketoexpressmygratitudetoallthosewho contributed,inonewayoranother,tothepreparationandimprovementofthebook. A.A.Borovkov
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