凯罗勒夫编著的《概率论和随机过程》是以作者在Princeton大学和Maryland大学的讲义为蓝本扩充而成，书中的内容正好可作为《概率论和随机过程》课程一学年的独立教材。这对于高年级的本科生、研究生和想要了解本科目基础知识的科研人员都是相当有用的。本书文笔流畅，其中的概念和相关的结果都是生动丰富，并具有启发性。每章末都包含难易不等的练习题。这本书已经被作者用作Princeton大学和Maryland高年级本科生和研究生学习该科目的一学期的教程。目次：（第一部分）概率论：随机变量及其分布；独立试验序列；勒贝格积分和数学期望；条件概率和期望；具有有限数状态的马尔科夫链；大数定理；测度的弱收敛；特征函数；极限定理；几个有趣的问题;（第二部分）随机过程：基本概念；条件期望和鞅；有限状态空间的马尔科夫链；广泛意义上的平稳随机过程；严格平稳随机过程；广义随机过程；布朗运动；马尔科夫过程和马尔科夫族；随机积分和Ito公式；随机微分方程；Gibbs随机域。

Part Ⅰ Probability Theory

 1 Random Variables and Their Distributions

1.1 Spaces of Elementary Outcomes， a-Algebras， and Measures

1.2 Expectation and Variance of Random Variables on a Discrete Probability Space

1.3 Probability of a Union of Events

1.4 Equivalent Formulations of a-Additivity， Borel a-Algebras and Measurability

1.5 Distribution Functions and Densities

1.6 Problems

 2 Sequences of Independent Trials

2.1 Law of Large Numbers and Applications

2.2 de Moivre-Laplace Limit Theorem and Applications

2.3 Poisson Limit Theorem.

2.4 Problems

 3 Lebesgue Integral and Mathematical Expectation

3.1 Definition of the Lebesgue Integral

3.2 Induced Measures and Distribution Functions

3.3 Types of Measures and Distribution Functions

3.4 Remarks on the Construction of the Lebesgue Measure

3.5 Convergence of Functions， Their Integrals， and the Fubini Theorem

3.6 Signed Measures and the R，adon-Nikodym Theorem

3.7 Lp Spaces

3.8 Monte Carlo Method

3.9 Problems

 4 Conditional Probabilities and Independence

4.1 Conditional Probabilities

4.2 Independence of Events， Algebras， and Random Variables

4.3

4.4 Problems

 5 Markov Chains with a Finite Number of States

5.1 Stochastic Matrices

5.2 Markov Chains

5.3 Ergodic and Non-Ergodic Markov Chains

5.4 Law of Large Numbers and the Entropy of a Markov Chain

5.5 Products of Positive Matrices

5.6 General Markov Chains and the Doeblin Condition

5.7 Problems

 6 Random Walks on the Lattice Zd

6.1 Recurrent and Transient R，andom Walks

6.2 Random Walk on Z and the Refiection Principle

6.3 Arcsine Law

6.4 Gambler's Ruin Problem

6.5 Problems

 7 Laws of Larze Numbers

7.1 Definitions， the Borel-Cantelli Lemmas， and the Kolmogorov Inequality

7.2 Kolmogorov Theorems on the Strong Law of Large Numbers

7.3 Problems

 8 Weak Converaence of Measures

8.1 Defnition of Weak Convergence

8.2 Weak Convergence and Distribution Functions

8.3 Weak Compactness， Tightness， and the Prokhorov Theorem

8.4 Problems

 9 Characteristic Functions

9.1 Definition and Basic Properties

9.2 Characteristic Functions and Weak Convergence

9.3 Gaussian Random Vectors

9.4 Problems

 10 Limit Theorems

10.1 Central Limit Theorem， the Lindeberg Condition

10.2 Local Limit Theorem

10.3 Central Limit Theorem and Renormalization GrOUD Theorv

10.4 Probabilities of Large Deviations

……

Part Ⅱ Random Processes

Index