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Graduate Texts in Mathematics Albert N. Shiryaev Probability-2 Third Edition Graduate Texts in Mathematics 95 Graduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: AlejandroAdem,UniversityofBritishColumbia DavidEisenbud,UniversityofCalifornia,Berkeley&MSRI BrianC.Hall,UniversityofNotreDame J.F.Jardine,UniversityofWesternOntario JeffreyC.Lagarias,UniversityofMichigan KenOno,EmoryUniversity JeremyQuastel,UniversityofToronto FadilSantosa,UniversityofMinnesota BarrySimon,CaliforniaInstituteofTechnology RaviVakil,StanfordUniversity StevenH.Weintraub,LehighUniversity GraduateTextsinMathematicsbridgethegapbetweenpassivestudyandcreative understanding, offering graduate-level introductions to advanced topics in mathe- matics.Thevolumesarecarefullywrittenasteachingaidsandhighlightcharacter- isticfeaturesofthetheory.Althoughthesebooksarefrequentlyusedastextbooks ingraduatecourses,theyarealsosuitableforindividualstudy. Moreinformationaboutthisseriesathttp://www.springer.com/series/136 Albert N. Shiryaev Probability-2 Third Edition † Translated by R.P. Boas and D.M. Chibisov 123 AlbertN.Shiryaev DepartmentofProbabilityTheory andMathematicalStatistics SteklovMathematicalInstituteand LomonosovMoscowStateUniversity Moscow,Russia TranslatedbyR.P.Boas†andD.M.Chibisov ISSN0072-5285 ISSN2197-5612 (electronic) GraduateTextsinMathematics ISBN978-0-387-72207-8 ISBN978-0-387-72208-5 (eBook) https://doi.org/10.1007/978-0-387-72208-5 LibraryofCongressControlNumber:2018953349 MathematicsSubjectClassification:60Axx,60Exx,60Fxx,60Gxx,60Jxx,62Lxx ©SpringerScience+BusinessMediaNewYork1984,1996 ©SpringerScience+BusinessMedia,LLC,partofSpringerNature2019 Originallypublishedinonevolume. TranslationfromtheRussianlanguageedition:Veroı(cid:2)atnost(cid:2)–2(fourthedition)byAlbertN.Shiryaev ©Shiryaev,A.N.2007and©MCCME2007.AllRightsReserved. Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerScience+BusinessMedia,LLC partofSpringerNature. Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A. PrefacetotheThirdEnglishEdition The present edition isatranslation of thefourthRussian edition of 2007, withthe previous three published in 1980, 1989, and 2004. The English translations of the firsttwoappearedin1984and1996.ThethirdandfourthRussianeditions,extended comparedtothesecondedition,werepublishedintwovolumestitledProbability-1 and Probability-2. Accordingly, the present edition consists of two volumes: this Vol.2, titled Probability-2, contains Chaps.4–8, and Chaps.1–3 are contained in Vol.1,titledProbability-1,whichwaspublishedin2016. ThisEnglishtranslationofProbability-2waspreparedbytheeditorandtransla- torProf.D.M.Chibisov,ProfessoroftheSteklovMathematicalInstitute.Aformer studentofN.V.Smirnov,hehasabroadviewofprobabilityandmathematicalstatis- tics,whichenabledhimnotonlytotranslatethepartsthathadnotbeentranslated before,butalsotoeditboththeprevioustranslationandtheRussiantext,makingin themquiteanumberofcorrectionsandamendments. TheauthorissincerelygratefultoD.M.Chibisovforthetranslationandscien- tificeditingofthisbook. Moscow,Russia A.Shiryaev 2018 PrefacetotheFourthRussianEdition A university course on probability and statistics usually consists of three one- semesterparts:probabilitytheory,randomprocesses,andmathematicalstatistics. The book Probability-1 covered the material normally included in probability theory. This book, Probability-2, contains extensive material for a course on random processes in the part dealing with discrete time processes, i.e., random sequences. (Thereaderinterestedinrandomprocesseswithcontinuoustimemayreferto[12], whichiscloselyrelatedtoProbability-1andProbability-2.) Chapter4,whichopensthisbook,isfocusedmostlyonthepropertiesofsumsof independentrandomvariablesthatholdwithprobabilityone(e.g.,“zero–one”laws, thestronglawoflargenumbers,thelawoftheiteratedlogarithm). Chapters5and6treatthestrictandwidesensestationaryrandomsequences. v vi In Chaps.7 and 8, we set out random sequences that form martingales and Markovchains.Theseclassesofprocessesenableustostudythebehaviorofvarious stochastic systems in the “future”, depending on their “past” and “present” thanks to which these processes play a very important role in modern probability theory anditsapplications. ThebookconcludeswithaHistoricalReviewoftheDevelopmentofMathemat- icalTheoryofProbability. Moscow,Russia A.Shiryaev 2003 Contents PrefacetotheThirdEnglishEdition................................. v PrefacetotheFourthRussianEdition ............................... v 4 SequencesandSumsofIndependentRandomVariables ........... 1 1 Zero–OneLaws........................................... 1 2 ConvergenceofSeries ..................................... 6 3 StrongLawofLargeNumbers............................... 12 4 LawoftheIteratedLogarithm ............................... 22 5 ProbabilitiesofLargeDeviations ............................ 27 5 Stationary(StrictSense)RandomSequences andErgodicTheory ........................................... 33 1 Stationary(StrictSense)RandomSequences:Measure-Preserving Transformations........................................... 33 2 ErgodicityandMixing ..................................... 37 3 ErgodicTheorems......................................... 39 6 Stationary(WideSense)RandomSequences:L2-Theory .......... 47 1 SpectralRepresentationoftheCovarianceFunction............. 47 2 OrthogonalStochasticMeasuresandStochasticIntegrals ........ 56 3 SpectralRepresentationofStationary(WideSense)Sequences ... 61 4 StatisticalEstimationofCovarianceFunction andSpectralDensity....................................... 71 5 Wold’sExpansion ......................................... 78 6 Extrapolation,Interpolation,andFiltering ..................... 85 7 TheKalman–BucyFilterandItsGeneralizations ............... 95 7 Martingales .................................................. 107 1 DefinitionsofMartingalesandRelatedConcepts ............... 107 2 PreservationofMartingalePropertyUnderaRandom TimeChange ............................................. 119 vii viii Contents 3 FundamentalInequalities ................................... 132 4 General Theorems on Convergence of Submartingales andMartingales........................................... 148 5 SetsofConvergenceofSubmartingalesandMartingales ......... 156 6 AbsoluteContinuityandSingularityofProbabilityDistributions onaMeasurableSpacewithFiltration ........................ 164 7 AsymptoticsoftheProbabilityoftheOutcomeofaRandomWalk withCurvilinearBoundary.................................. 178 8 CentralLimitTheoremforSumsofDependent RandomVariables ......................................... 183 9 DiscreteVersionofItoˆ’sFormula ............................ 197 10 ApplicationofMartingaleMethodstoCalculationofProbability ofRuininInsurance ....................................... 202 11 FundamentalTheoremsofStochasticFinancialMathematics:The MartingaleCharacterizationoftheAbsenceofArbitrage......... 207 12 HedginginArbitrage-FreeModels ........................... 220 13 OptimalStoppingProblems:MartingaleApproach.............. 228 8 MarkovChains ............................................... 237 1 DefinitionsandBasicProperties ............................. 237 2 GeneralizedMarkovandStrongMarkovProperties ............. 249 3 Limiting,Ergodic,andStationaryProbabilityDistributions forMarkovChains ........................................ 256 4 ClassificationofStatesofMarkovChainsinTermsofAlgebraic PropertiesofMatricesofTransitionProbabilities ............... 259 5 ClassificationofStatesofMarkovChainsinTermsofAsymptotic PropertiesofTransitionProbabilities ......................... 265 6 Limiting,Stationary,andErgodicDistributionsforCountable MarkovChains ........................................... 277 7 Limiting,Stationary,andErgodicDistributionsforFiniteMarkov Chains................................................... 283 8 SimpleRandomWalkasaMarkovChain ..................... 284 9 OptimalStoppingProblemsforMarkovChains ................ 296 DevelopmentofMathematicalTheoryofProbability: HistoricalReview.................................................. 313 HistoricalandBibliographicalNotes(Chaps.4–8)..................... 333 References........................................................ 339 Index ............................................................ 343 Table of Contents of Probability-1 PrefacetotheThirdEnglishEdition PrefacetotheFourthRussianEdition PrefacetotheThirdRussianEdition PrefacetotheSecondEdition PrefacetotheFirstEdition Introduction 1 ElementaryProbabilityTheory 1 ProbabilisticModelofanExperimentwithaFiniteNumberofOutcomes 2 SomeClassicalModelsandDistributions 3 ConditionalProbability:Independence 4 RandomVariablesandTheirProperties 5 TheBernoulliScheme:I—TheLawofLargeNumbers 6 The Bernoulli Scheme: II—Limit Theorems (Local, de Moivre–Laplace, Poisson) 7 EstimatingtheProbabilityofSuccessintheBernoulliScheme 8 Conditional Probabilities and Expectations with Respect to Decomposi- tions 9 RandomWalk:I—ProbabilitiesofRuinandMeanDurationinCoinToss- ing 10 RandomWalk:II—ReflectionPrinciple—ArcsineLaw 11 Martingales:SomeApplicationstotheRandomWalk 12 MarkovChains:ErgodicTheorem,StrongMarkovProperty 13 GeneratingFunctions 14 Inclusion–ExclusionPrinciple 2 MathematicalFoundationsofProbabilityTheory 1 Kolmogorov’sAxioms 2 Algebrasandσ-Algebras:MeasurableSpaces ix
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