概率论（英文版）

　　In the Preface to the first edition， originally published in 1980， we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow， which were issued， in part， in mimeographed form under the title "Probability， Statistics， and Stochastic Processors， I， II" and published by that University. Our original intention in writing the first edition of this book was to divide the contents into three parts： probability， mathematical statistics， and theory of stochastic processes， which corresponds to an outline of a threesemester course of lectures for university students of mathematics. However， in the course of preparing the book， it turned out to be impossible to realize this intention completely， since a full exposition would have required too much space. In this connection， we stated in the Preface to the first edition that only probability theory and the theory of random processes with discrete time were really adequately presented.

暂缺《概率论（英文版）》作者简介

PrefacetotheSecondEdition
PrefacetotheFirstEdition
Introduction
CHAPTERIElementaryProbabilityTheory
1.ProbabilisticModelofanExperimentwithaFiniteNumberofOutcomes
2.SomeClassicalModelsandDistributions
3.ConditionalProbability.Independence
4.RandomVariablesandTheirProperties
5.TheBernoulliScheme.I.TheLawofLargeNumbers
6.TheBernoulliScheme.II.LimitTheorems(Local,DeMoivre-Laplace,Poisson)
7.EstimatingtheProbabilityofSuccessintheBernoulliScheme
8.ConditionalProbabilitiesandMathematicalExpectationswithRespecttoDecompositions
9.RandomWalk.I.ProbabilitiesofRuinandMeanDurationinCoinTossing
10.RandomWalk.II.ReflectionPrinciple.ArcsineLaw
11.Martingales.SomeApplicationstotheRandomWalk
12.MarkovChains.ErgodicTheorem.StrongMarkovProperty
CHAPTERIIMathematicalFoundationsofProbabilityTheory
1.ProbabilisticModelforanExperimentwithInfinitelyManyOutcomes.Kolmogorov'sAxioms
2.Algebrasanda-algebras.MeasurableSpaces
3.MethodsofIntroducingProbabilityMeasuresonMeasurableSpaces
4.RandomVariables.I.
5.RandomElements
6.LebesgueIntegral.Expectation
7.ConditionalProbabilitiesandConditionalExpectationswithRespecttoaa-Algebra
8.RandomVariables.II.
9.ConstructionofaProcesswithGivenFinite-DimensionalDistribution
10.VariousKindsofConvergenceofSequencesofRandomVariables
11.TheHilbertSpaceofRandomVariableswithFiniteSecondMoment
12.CharacteristicFunctions
13.GaussianSystems
CHAPTERIIIConvergenceofProbabilityMeasures.CentralLimitTheorem
1.WeakConvergenceofProbabilityMeasuresandDistributions
2.RelativeCompactnessandTightnessofFamiliesofProbabilityDistributions
3.ProofsofLimitTheoremsbytheMethodofCharacteristicFunctions
4.CentralLimitTheoremforSumsofIndependentRandomVariables.I.TheLindebergCondition
5.CentralLimitTheoremforSumsofIndependentRandomVariables.II.NonclassicalConditions
6.InfinitelyDivisibleandStableDistributions
7.MetrizabilityofWeakConvergence
8.OntheConnectionofWeakConvergenceofMeasureswithAlmostSureConvergenceofRandomElements("MethodofaSingleProbabilitySpace")
9.TheDistanceinVariationbetweenProbabilityMeasures.
Kakutani-HellingerDistanceandHellingerIntegrals.ApplicationtoAbsoluteContinuityandSingularityofMeasures
10.ContiguityandEntireAsymptoticSeparationofProbabilityMeasures
11.RapidityofConvergenceintheCentralLimitTheorem
12.RapidityofConvergenceinPoisson'sTheorem
CHAPTERIVSequencesandSumsofIndependentRandomVariables
1.Zero-or-OneLaws
2.ConvergenceofSeries
3.StrongLawofLargeNumbers
4.LawoftheIteratedLogarithm
5.RapidityofConvergenceintheStrongLawofLargeNumbersandintheProbabilitiesofLargeDeviations
CHAPTERV
Stationary(StrictSense)RandomSequencesandErgodicTheory
1.Stationary(StrictSense)RandomSequences.Measure-PreservingTransformations
2.ErgodicityandMixing
3.ErgodicTheorems
CHAPTERVI
Stationary(WideSense)RandomSequences.L2Theory
1.SpectralRepresentationoftheCovarianceFunction
2.OrthogonalStochasticMeasuresandStochasticIntegrals
3.SpectralRepresentationofStationary(WideSense)Sequences
4.StatisticalEstimationoftheCovarianceFunctionandtheSpectralDensity
5.Wold'sExpansion
6.Extrapolation.InterpolationandFiltering
7.TheKalman-BucyFilterandItsGeneralizations
CHAPTERVII
SequencesofRandomVariablesthatFormMartingales
1.DefinitionsofMartingalesandRelatedConcepts
2.PreservationoftheMartingalePropertyUnderTimeChangeataRandomTime
3.FundamentalInequalities
4.GeneralTheoremsontheConvergenceofSubmartingalesandMartingales
5.SetsofConvergenceofSubmartingalesandMartingales
6.AbsoluteContinuityandSingularityofProbabilityDistributions
7.AsymptoticsoftheProbabilityoftheOutcomeofaRandomWalkwithCurvilinearBoundary
8.CentralLimitTheoremforSumsofDependentRandomVariables
9.DiscreteVersionofIto'sFormula
10.ApplicationstoCalculationsoftheProbabilityofRuininInsurance
CHAPTERVIII
SequencesofRandomVariablesthatFormMarkovChains
1.DefinitionsandBasicProperties
2.ClassificationoftheStatesofaMarkovChaininTermsofArithmeticPropertiesoftheTransitionProbabilitiesPij(n)
3.ClassificationoftheStatesofaMarkovChaininTermsofAsymptoticPropertiesoftheProbabilitiesPij(n)
4.OntheExistenceofLimitsandofStationaryDistributions
5.Examples
HistoricalandBibliographicalNotes
References
InclexofSymbols
Index