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Universitext Achim Klenke Probability Theory AComprehensiveCourse 123 Prof.Dr.AchimKlenke Institutfu¨rMathematik JohannesGutenberg-Universita¨tMainz Staudingerweg9 55099Mainz Germany ISBN:978-1-84800-047-6 e-ISBN:978-1-84800-048-3 DOI:10.1007/978-1-84800-048-3 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2007939558 MathematicsSubjectClassification(2000):60-01;28-01 TranslationfromtheGermanlanguageedition: WahrscheinlichkeitstheoriebyAchimKlenke Copyright(cid:1)cSpringerVerlagBerlinHeidelberg2006 SpringerisapartofSpringerScience+BusinessMedia AllRightsReserved (cid:1)cSpringer-VerlagLondonLimited2008 Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview, aspermittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybe reproduced,storedortransmitted,inanyformorbyanymeans,withthepriorpermissionin writingofthepublishers,orinthecaseofreprographicreproductioninaccordancewiththe termsoflicensesissuedbytheCopyrightLicensingAgencies.Enquiriesconcerningrepro- ductionoutsidethosetermsshouldbesenttothepublishers. Theuseofregisteredname,trademarks,etc.,inthispublicationdoesnotimply,eveninthe absenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantlawsandregu- lationsandthereforefreeforgeneraluse. Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyofthe informationcontainedinthisbookandcannotacceptanylegalresponsibilityorliabilityfor anyerrorsoromissionsthatmaybemade. Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 SpringerScience+BusinessMedia springer.com Preface This book is based on two four-hour courses on advanced probability theory that I haveheldinrecentyearsattheuniversitiesofCologneandMainz.Itisimplicitlyas- sumedthatthereaderhasacertainfamiliaritywiththebasicconceptsofprobability theory,althoughtheformalframeworkwillbefullydevelopedinthisbook. The aim of this book is to present the central objects and concepts of probability theory: random variables, independence, laws of large numbers and central limit theorems, martingales, exchangeability and infinite divisibility, Markov chains and Markov processes, as well as their connection with discrete potential theory, cou- pling, ergodic theory, Brownian motion and the Itoˆ integral (including stochastic differentialequations),thePoissonpointprocess,percolationandthetheoryoflarge deviations. Measure theory and integration are necessary prerequisites for a systematic proba- bilitytheory.Wedevelopitonlytothepointtowhichitisneededforourpurposes: construction of measures and integrals, the Radon-Nikodym theorem and regular conditional distributions,convergence theoremsforfunctions(Lebesgue)andmea- sures (Prohorov) and construction of measures in product spaces. The chapters on measure theory do not come as a block at the beginning (although they are written such that this would be possible; that is, independent of the probabilistic chapters) butareratherinterlacedwithprobabilisticchaptersthataredesignedtodisplaythe poweroftheabstractconceptsinthemoreintuitiveworldofprobabilitytheory.For example, we study percolation theory at the point where we barely have measures, randomvariablesandindependence;noteventheintegralisneeded.Astheonlyex- ception,thesystematicconstructionofindependentrandomvariablesisdeferredto Chapter 14. Although it is rather a matter of taste, I hope that this setup helps to motivatethereaderthroughoutthemeasure-theoreticalchapters. Those readers with a solid measure-theoretical education can skip in particular the firstandfourthchaptersandmightwishonlytolookupthisorthat. VI Preface Inthefirsteightchapters,welaythefoundationsthatwillbeneededinallsubsequent chapters. After that, there are seven more or less independent parts, consisting of Chapters9–12,13,14,15–16,17–19,20and23.ThechapteronBrownianmotion (21)makesreferencetoChapters9–15.Again,afterthat,thethreeblocksconsisting ofChapters22,24and25–26canbereadindependently. I should like to thank all those who read the manuscript and the German original versionofthisbookandgavenumeroushintsforimprovements:RolandAlkemper, Rene´ Billing, Dirk Bru¨ggemann, Anne Eisenbu¨rger, Patrick Jahn, Arnulf Jentzen, Ortwin Lorenz, L. Mayer, Mario Oeler, Marcus Scho¨lpen, my colleagues Ehrhard Behrends, Wolfgang Bu¨hler, Nina Gantert, Rudolf Gru¨bel, Wolfgang Ko¨nig, Peter Mo¨rters and Ralph Neininger, and in particular my colleague from Munich Hans- OttoGeorgii.DrJohnPreaterdidagreatjoblanguageeditingtheEnglishmanuscript andalsopointingoutnumerousmathematicalflaws. IamespeciallyindebtedtomywifeKatrinforproofreadingtheEnglishmanuscript andforherpatienceandsupport. Iwouldbegratefulforfurthersuggestions,errorsetc.tobesentbye-mailto [email protected] Mainz, AchimKlenke October2007 Contents Preface............................................................ V 1 BasicMeasureTheory .......................................... 1 1.1 ClassesofSets............................................. 1 1.2 SetFunctions .............................................. 12 1.3 TheMeasureExtensionTheorem ............................. 18 1.4 MeasurableMaps .......................................... 34 1.5 RandomVariables .......................................... 43 2 Independence.................................................. 49 2.1 IndependenceofEvents ..................................... 49 2.2 IndependentRandomVariables ............................... 56 2.3 Kolmogorov’s0-1Law...................................... 63 2.4 Example:Percolation ....................................... 66 3 GeneratingFunctions........................................... 77 3.1 DefinitionandExamples .................................... 77 3.2 PoissonApproximation ..................................... 80 3.3 BranchingProcesses ........................................ 82 4 TheIntegral ................................................... 85 4.1 ConstructionandSimpleProperties ........................... 85 4.2 MonotoneConvergenceandFatou’sLemma .................... 93 4.3 LebesgueIntegralversusRiemannIntegral ..................... 95 VIII Contents 5 MomentsandLawsofLargeNumbers ........................... 101 5.1 Moments ................................................. 101 5.2 WeakLawofLargeNumbers ................................ 108 5.3 StrongLawofLargeNumbers................................ 111 5.4 SpeedofConvergenceintheStrongLLN ...................... 119 5.5 ThePoissonProcess ........................................ 123 6 ConvergenceTheorems ......................................... 129 6.1 AlmostSureandMeasureConvergence ........................ 129 6.2 UniformIntegrability ....................................... 134 6.3 ExchangingIntegralandDifferentiation........................ 140 7 Lp-SpacesandtheRadon-NikodymTheorem ..................... 143 7.1 Definitions ................................................ 143 7.2 InequalitiesandtheFischer-RieszTheorem..................... 145 7.3 HilbertSpaces ............................................. 151 7.4 Lebesgue’sDecompositionTheorem .......................... 154 7.5 Supplement:SignedMeasures................................ 158 7.6 Supplement:DualSpaces.................................... 165 8 ConditionalExpectations ....................................... 169 8.1 ElementaryConditionalProbabilities .......................... 169 8.2 ConditionalExpectations .................................... 173 8.3 RegularConditionalDistribution.............................. 179 9 Martingales ................................................... 189 9.1 Processes,Filtrations,StoppingTimes ......................... 189 9.2 Martingales ............................................... 194 9.3 DiscreteStochasticIntegral .................................. 198 9.4 DiscreteMartingaleRepresentationTheoremandtheCRRModel.. 200 10 OptionalSamplingTheorems.................................... 205 10.1 DoobDecompositionandSquareVariation ..................... 205 10.2 OptionalSamplingandOptionalStopping...................... 209 Contents IX 10.3 UniformIntegrabilityandOptionalSampling ................... 214 11 MartingaleConvergenceTheoremsandTheirApplications ......... 217 11.1 Doob’sInequality .......................................... 217 11.2 MartingaleConvergenceTheorems............................ 219 11.3 Example:BranchingProcess ................................. 228 12 BackwardsMartingalesandExchangeability...................... 231 12.1 ExchangeableFamiliesofRandomVariables ................... 231 12.2 BackwardsMartingales ..................................... 236 12.3 DeFinetti’sTheorem ....................................... 239 13 ConvergenceofMeasures ....................................... 245 13.1 ATopologyPrimer ......................................... 245 13.2 WeakandVagueConvergence................................ 251 13.3 Prohorov’sTheorem ........................................ 259 13.4 Application:AFreshLookatdeFinetti’sTheorem .............. 268 14 ProbabilityMeasuresonProductSpaces.......................... 271 14.1 ProductSpaces............................................. 272 14.2 FiniteProductsandTransitionKernels......................... 275 14.3 Kolmogorov’sExtensionTheorem ............................ 283 14.4 MarkovSemigroups ........................................ 288 15 CharacteristicFunctionsandtheCentralLimitTheorem........... 293 15.1 SeparatingClassesofFunctions .............................. 293 15.2 CharacteristicFunctions:Examples ........................... 300 15.3 Le´vy’sContinuityTheorem .................................. 307 15.4 CharacteristicFunctionsandMoments......................... 312 15.5 TheCentralLimitTheorem .................................. 317 15.6 MultidimensionalCentralLimitTheorem ...................... 324 16 InfinitelyDivisibleDistributions ................................. 327 16.1 Le´vy-KhinchinFormula..................................... 327 X Contents 16.2 StableDistributions......................................... 339 17 MarkovChains ................................................ 345 17.1 DefinitionsandConstruction ................................. 345 17.2 DiscreteMarkovChains:Examples ........................... 352 17.3 DiscreteMarkovProcessesinContinuousTime ................. 356 17.4 DiscreteMarkovChains:RecurrenceandTransience............. 361 17.5 Application:RecurrenceandTransienceofRandomWalks........ 365 17.6 InvariantDistributions ...................................... 372 18 ConvergenceofMarkovChains.................................. 379 18.1 PeriodicityofMarkovChains ................................ 379 18.2 CouplingandConvergenceTheorem .......................... 383 18.3 MarkovChainMonteCarloMethod........................... 390 18.4 SpeedofConvergence ...................................... 398 19 MarkovChainsandElectricalNetworks.......................... 403 19.1 HarmonicFunctions ........................................ 404 19.2 ReversibleMarkovChains ................................... 407 19.3 FiniteElectricalNetworks ................................... 408 19.4 RecurrenceandTransience................................... 414 19.5 NetworkReduction......................................... 421 19.6 RandomWalkinaRandomEnvironment....................... 427 20 ErgodicTheory ................................................ 431 20.1 Definitions ................................................ 431 20.2 ErgodicTheorems.......................................... 435 20.3 Examples ................................................. 437 20.4 Application:RecurrenceofRandomWalks ..................... 439 20.5 Mixing ................................................... 442 21 BrownianMotion .............................................. 447 21.1 ContinuousVersions ........................................ 447 21.2 ConstructionandPathProperties.............................. 454 Contents XI 21.3 StrongMarkovProperty..................................... 459 21.4 Supplement:FellerProcesses................................. 462 21.5 ConstructionviaL2-Approximation........................... 465 21.6 TheSpaceC([0,∞)) ....................................... 469 21.7 ConvergenceofProbabilityMeasuresonC([0,∞)).............. 471 21.8 Donsker’sTheorem......................................... 474 21.9 PathwiseConvergenceofBranchingProcesses∗ ................. 477 21.10SquareVariationandLocalMartingales ....................... 483 22 LawoftheIteratedLogarithm................................... 495 22.1 IteratedLogarithmfortheBrownianMotion.................... 495 22.2 Skorohod’sEmbeddingTheorem ............................. 498 22.3 Hartman-WintnerTheorem .................................. 503 23 LargeDeviations............................................... 505 23.1 Crame´r’sTheorem.......................................... 506 23.2 LargeDeviationsPrinciple................................... 510 23.3 Sanov’sTheorem........................................... 514 23.4 Varadhan’sLemmaandFreeEnergy........................... 519 24 ThePoissonPointProcess....................................... 525 24.1 RandomMeasures.......................................... 525 24.2 PropertiesofthePoissonPointProcess ........................ 529 24.3 ThePoisson-DirichletDistribution∗ ........................... 535 25 TheItoˆ Integral................................................ 543 25.1 Itoˆ IntegralwithRespecttoBrownianMotion................... 543 25.2 Itoˆ IntegralwithRespecttoDiffusions ......................... 551 25.3 TheItoˆ Formula............................................ 554 25.4 DirichletProblemandBrownianMotion ....................... 562 25.5 RecurrenceandTransienceofBrownianMotion................. 564 26 StochasticDifferentialEquations ................................ 567 26.1 StrongSolutions ........................................... 567