概率论教程

Index

Preface to the third editioniii

Preface to the second editionv

Preface to the first editionvii

1 Distribution function

1.1 Monotone functionsl

1.2 Distribution functions

1.3 Absolutely continuous and singular distributions

2 Measure theory

2.1 Classes of sets

2.2 Probability measures and their distribution functions

3 Random variable. Expectation. Independence

3.1 General definitions

3.2 Properties of mathematical expectation

3.3 Independence

4 Convergence concepts

4.1 Various modes of convergence

4.2 Almost sure convergence; Borel-Cantelli lemma

4.3 Vague convergence

4.4 Continuation

4.5 Uniform integrability; convergence of moments

5 Law of large numbers. Random series

5.1 Simple limit theorems

5.2 Weak law of large numbers

5.3 Convergence of series

5.4 Strong law of large numbers

5.5 Applications

Bibliographical Note

6 Characteristic function

6.1 General properties; convolutions

6.2 Uniqueness and inversion

6.3 Convergence theorems

6.4 Simple applications

6.5 Representation theorems

6.6 Multidimensional case; Laplace transforms

Bibliographical Note

7 Central limit theorem and its ramifications

7.1 Liapounovs theorem

7.2 Lindeberg-FeUer theorem

7.3 Ramifications of the central limit theorem

7.4 Error estimation

7.5 Law of the iterated logarithm

7.6 Infinite divisibility

Bibliographical Note

8 Random walk

8.1 Zero-or-one laws

8.2 Basic notions

8.3 Recurrence

8.4 Fine structure

8.5 Continuation

Bibliographical Note

9 Conditioning. Markov property. Martingale

9.1 Basic properties of conditional expectation3 l

9.2 Conditional independence; Markov property

9.3 Basic properties of smartingales

9.4 Inequalities and convergence

9.5 Applications

Bibliographical Note

Supplement: Measure and Integral

1 Construction of measure

2 Characterization of extensions

3 Measures in R

4 Integral

5 Applications

General Bibliography

随机变量和分布函数，测度论，数学期望，方差，各种收敛性，大数律， 中心极限定理，特征函数，随机游动， 马氏性和鞅理论.本书内容丰富，逻辑紧密，叙述严谨，不仅可以扩展读者的视野，而且还将为其后续的学习和研究打下坚实基础。此外，本书的习题较多， 都经过细心的遴选， 从易到难， 便于读者巩固练习。本版补充了有关测度和积分方面的内容，并增加了一些习题。
本书是一本享誉世界的经典概率论教材，令众多读者受益无穷，自出版以来，已被世界75%以上的大学的数万名学生使用。本书内容丰富，逻辑清晰，叙述严谨，不仅可以拓展读者的视野，而且还将为其后续的学习和研究打下坚实基础。此外，本书的习题较多， 都经过细心的遴选, 从易到难, 便于读者巩固练习。本版补充了有关测度和积分方面的内容，并增加了一些习题。