Probability theory: a comprehensive course

620 Pages·2008·3.213 MB·English

by  Prof. Dr. Achim Klenke (auth.)

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
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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