这是一套公认的概率论经典教科书，可供高年级大学生和研究生使用，同时也是概率论和统计学方面的专家，学者经常使用的参考书。在这套书的第4版中增加了距离空间测定，随机游动，布朗运动及不变原理，后两部门尤为精彩。

本书为其中一册。

Preface to the Second Edition

Preface to the First Edition

Introduction

CHAPTER I

Elementary Probability Theory

1. Probabilistic Model ofan Experiment with a Finite Number of Outcomes

2. Some Classical Models and Distributions

3. Conditional Probability. Independence

4. Random Variables and Their Properties

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

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

7. Estimating the Probability of Success in the Bernoulli Scheme

8. Conditional Probabilities and Mathematical Expectations with Respect to Decompositions

9. Random Walk. I. Probabilities of Ruin and Mean Duration in Coin Tossing

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

11. Martingales. SomeApplications to the Random Walk

12. Markov Chains. Ergodic Theorem. Strong Markov Property

CHAPTER II

MathematiCal Foundations of Probability Theory

1. Probabilistic Model for an Experiment with Infinitely Many Outcomes. Kolmogorov's Axioms

2. Algebras and-algebras. Measurable Spaces

3. Methods of Introducing Probability Measures on Measurable Spaces

4. Random Variables. I

5. Random Elements

6. Lebesgue Integral. Expectation

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

8. Random Variables. II

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

10. Various Kinds of Convergence of Sequences of Random Variables

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

12. Characteristic Functions

13. Gaussian Systems

CHAPTER III

Convergence of Probability Measures. Central Limit

Theorem

1. Weak Convergence of Probability Measures and Distributions

2. Relative Compactness and Tightness of Families of Probability Distributions

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

4. Central Limit Theorem for Sums of Independent Random Variables. I. The Lindeberg Condition

5. Central Limit Theorem for Sums of Independent Random Variables. II. Nonclassical Conditions

6. Infinitely Divisible and Stable Distributions

7. Metrizability of Weak Convergence

8. On the Connection of Weak Convergence of Measures with Almost Sure Convergence of Random Elements ("Method of a Single Probability Space")

9. The Distance in Variation between Probability Measures.

 Kakutani-Hellinger Distance and Hdlinger Integrals. Application to

 Absolute Continuity and Singularity of Measures

10. Contiguity and Entire Asymptotic Separation of Probability Measures

11. Rapidity of Convergence in the Central Limit Theorem

12. Rapidity of Convergence in Poisson's Theorem

CHAPTER IV

Sequences and Sums of Independent Random Variables

1. Zero-or-One Laws

2. Convergence of Series

3. Strong Law of Large Numbers

4. Law of the Iterated Logarithm

5. Rapidity of Convergence in the Strong Law of Large Numbers and in the Probabilities of Large Deviations

CHAPTER V

Stationary (Strict Sense) Random Sequences and

Ergodic Theory

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

2. Ergodicity and Mixing

3. Ergodic Theorems

CHAPTER VI

Stationary (Wide Sense) Random Sequences. L2 Theory

l. Spectral Representation of the Covariance Function

2. Orthogonal Stochastic Measures and Stochastic Integrals

3. Spectral Representation of Stationary (Wide Sense) Sequences

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

5. Wold's Expansion

6. Extrapolation. Interpolation and Filtering

7. The Kalman-Bucy Filter and Its Generalizations

CHAPTER VII

Sequences of Random Variables that Form Martingales

1. Definitions of Martingales and Related Concepts

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

3. Fundamental Inequalities

4. General Theorems on the Convergence of Submartingales and Martingales

5. Sets of Convergence of Submartingales and Martingales

6. Absolute Continuity and Singularity of Probability Distributions

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

8. Central Limit Theorem for Sums of Dependent Random Variables

9. Discrete Version of It6's Formula

10. Applications to Calculations of the Probability of Ruin in Insurance

CHAPTER VIII

Sequences of Random Variables that Form Markov Chains

1. Definitions and Basic Properties

2. Classification of the States of a Markov Chain in Terms of Arithmetic Properties of the Transition Probabilities p])

3. Classification of the States of a Markov Chain in Terms of Asymptotic Properties of the Probabilities pl')

4. On the Existence of Limits and of Stationary Distributions

5. Examples

Historical and Bibliographical Notes

References

Index of Symbols

Index