概率论(第2版)(英文版)

本书是一部经典的概率论研究生教材。本书的 俄文原版于1979年出版，作者在莫斯科大学曾以本 书内容为蓝本，为概率论研究生讲课多年，教学效 果很好。1984年Springer出版了该书的英文版，本 书是该英文版的第2版，本版内容比第1版增加了30 余页。 目次：初等概率论；概率论的数学基础；概率 测度的收敛性及中心极限定理；独立随机变量的序 列与加项；平稳随机序列及遍历性理论；平稳随机 序列及L2理论；形成鞅的随机变量序列；形成马尔 可夫链的随机变量序列。 读者对象：数学专业的研究生。

Preface to the Second Edition Preface to the First Edition Introduction CHAPTER Ⅰ Elementary Probability Theory §1.Probabilistic Model of an 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-Laplace, 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. Some Applications 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 o-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 o-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 Hellinger 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