概率论和随机过程 : 第2版

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

This book is primarily based on a one-year course that has been taught for a number of years at Princeton University to advanced undergraduate and graduate students. During the last year a similar course has also been taught at the University of Maryland.
We would like to express our thanks to Ms. Sophie Lucas and Prof. Rafael Herrera who read the manuscript and suggested many corrections. We are particularly grateful to Prof. Boris Gurevich for making many important sug-gestions on both the mathematical content and style.
While writing this book， L. Koralov was supported by a National Sci-ence Foundation grant （DMS-0405152）. Y. Sinai was supported by a National Science Foundation grant （DMS-0600996）.