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Introduction 1

CHAPTER Ⅰ Elementary Probability Theory 5

1.Probabilistic Model of an Experiment with a Finite Number of Outcomes 5

2.Some Classical Models and Distributions 17

3.Conditional Probability.Independence 23

4.Random Variables and Their Properties 32

5.The Bernoulli Scheme.Ⅰ.The Law of Large Numbers 45

6.The Bernoulli Scheme.Ⅱ.Limit Theorems （Local,De Moivre-Laplace, Poisson） 55

7.Estimating the Probability of Success in the Bernoulli Scheme 68

8.Conditional Probabilities and Mathematical Expectations withRespect to Decompositions 74

9.Random Walk.I.Probabilities of Ruin and Mean Duration inCoin Tossing 81

10.Random Walk.Ⅱ.Reflection Principle.Arcsine Law 92

11.Martingales.Some Applications to the Random Walk 101

12.Markov Chains.Ergodic Theorem.Strong Markov Property 108

CHAPTER Ⅱ Mathematical Foundations of Probability Theory 129

1.Probabilistic Model for an Experiment with Infinitely ManyOutcomes.Kolmogorov’s Axioms 129

2.Algebras and σ-algebras.Measurable Spaces 137

3.Methods of Introducing Probability Measures on Measurable Spaces 149

4.Random Variables.Ⅰ 164

5.Random Elements 174

6.Lebesgue Integral Expectation 178

7.Conditional Probabilities and Conditional Expectations with Respect to a σ-Algebra 210

8.Random Variables.Ⅱ 232

9.Construction of a Process with Given Finite-Dimensional Distribution 243

10.Various Kinds of Convergence of Sequences of Random Variables 250

11.The Hilbert Space of Random Variables with Finite Second Moment 260

12.Characteristic Functions 272

13.Gaussian Systems 295

CHAPTER Ⅲ Convergence of Probability Measures.Central Limit Theorem 306

1.Weak Convergence of Probability Measures and Distributions 306

2.Relative Compactness and Tightness of Families of Probability Distributions 314

3.Proofs of Limit Theorems by the Method of Characteristic Functions 318

4.Central Limit Theorem for Sums of Independent Random Variables 326

5.Infinitely Divisible and Stable Distributions 335

6.Rapidity of Convergence in the Central Limit Theorem 342

7.Rapidity of Convergence in Poisson’s Theorem 345

CHAPTER Ⅳ Sequences and Sums of Independent Random Variables 354

1.Zero-or-One Laws 354

2.Convergence of Series 359

3.Strong Law of Large Numbers 363

4.Law of the Iterated Logarithm 370

CHAPTER Ⅴ Stationary （Strict Sense） Random Sequences and Ergodic Theory 376

1.Stationary （Strict Sense） Random Sequences.Measure-Preserving Transformations 376

2.Ergodicity and Mixing 379

3.Ergodic Theorems 381

CHAPTER Ⅵ Stationary （Wide Sense） Random Sequences.L2 Theory 387

1.Spectral Representation of the Covariance Function 387

2.Orthogonal Stochastic Measures and Stochastic Integrals 395

3.Spectral Representation of Stationary （Wide Sense） Sequences 401

4.Statistical Estimation of the Covariance Function and the Spectral Density 412

5.Wold’s Expansion 418

6.Extrapolation, Interpolation and Filtering 425

7.The Kalman-Bucy Filter and Its Generalizations 436

CHAPTER Ⅶ Sequences of Random Variables that Form Martingales 446

1.Definitions of Martingales and Related Concepts 446

2.Preservation of the Martingale Property Under Time Change at a Random Time 456

3.Fundamental Inequalities 464

4.General Theorems on the Convergence of Submartingales and Martingales 476

5.Sets of Convergence of Submartingales and Martingales 483

6.Absolute Continuity and Singularity of Probability Distributions 492

7.Asymptotics of the Probability of the Outcome of a Random Walk with Curvilinear Boundary 504

8.Central Limit Theorem for Sums of Dependent Random Variables 509

CHAPTER Ⅷ Sequences of Random Variables that Form Markov Chains 523

1.Definitions and Basic Properties 523

2.Classification of the States of a Markov Chain in Terms of Arithmetic Properties of the Transition Probabilities p（n）ij 528

3.Classification of the States of a Markov Chain in Terms of Asymptotic Properties of the Probabilities p（n）ij 532

4.On the Existence of Limits and of Stationary Distributions 541

5.Examples 546

Historical and Bibliographical Notes 555

References 561

Index of Symbols 565

Index 569