Table Of ContentGraduate Texts in Mathematics
Albert N. Shiryaev
Probability-2
Third Edition
Graduate Texts in Mathematics 95
Graduate Texts in Mathematics
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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
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©SpringerScience+BusinessMedia,LLC,partofSpringerNature2019
Originallypublishedinonevolume.
TranslationfromtheRussianlanguageedition:Veroı(cid:2)atnost(cid:2)–2(fourthedition)byAlbertN.Shiryaev
©Shiryaev,A.N.2007and©MCCME2007.AllRightsReserved.
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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
