【最受欢迎的概率书】《概率论：理论与实例》，490页pdf

2020 年 11 月 13 日 专知

Rick Durrett，北卡罗来纳州杜克大学数学系教授并获得了博士学位。1976年获得斯坦福大学运筹学博士学位。在加利福尼亚大学洛杉矶分校工作了9年，在康奈尔大学工作了25年后，他于2010年移居杜克大学。他撰写了8本著作和220余篇期刊文章，主题广泛，并指导了超过45位博士。他是美国国家科学院、美国艺术与科学学院的成员，以及数学统计研究所和美国数学学会的会员。

《概率论：理论与实例》，包含优秀的例子与有趣（又富有挑战性）的习题很好地协助了理论的延伸。这些例子也许并没有被完全应用于实际，但却清楚地解释了前面的理论。而且，是的，仅靠前面的理论与例题不足以解决所有习题。你需要想更多，解决问题也许需要用到之前的习题——但这不正是真正的数学研究的样子吗？你永远不会知道从哪能找到答案。

https://www.cambridge.org/hk/academic/subjects/statistics-probability/probability-theory-and-stochastic-processes/probability-theory-and-examples-5th-edition?format=HB&isbn=9781108473682

目录：

1. Measure Theory

1. Probability Spaces
2. Distributions
3. Random Variables
4. Integration
5. Properties of the Integral
6. Expected Value
7. Product Measures, Fubini's Theorem

2. Laws of Large Numbers

1. Independence
2. Weak Laws of Large Numbers
3. Borel-Cantelli Lemmas
4. Strong Law of Large Numbers
5. Convergence of Random Series*
6. Renewal Theory*
7. Large Deviations*

3. Central Limit Theorems

1. The De Moivre-Laplace Theorem
2. Weak Convergence
3. Characteristic Functions
4. Central Limit Theorems
5. Local Limit Theorems*
6. Poisson Convergence
7. Poisson Processes
8. Stable Laws*
9. Infinitely Divisible Distributions*
10. Limit Theorems in R^d *

4. Martingales

1. Conditional Expectation
2. Martingales, Almost Sure Convergence
3. Examples
4. Doob's Inequality, L^p Convergence
5. Square Integrable Martingales (was Subsection 5.4.1)
6. Uniform Integrability, Convergence in L¹
7. Backwards Martingales
8. Optional Stopping Theorems
9. Combinatorics of Simple Random Walk

5. Markov Chains

1. Examples
2. Construction, Markov Properties
3. Recurrence and Transience
4. Recurrence of Random Walks
5. Stationary Measures
6. Asymptotic Behavior
7. Periodicity, Tail σ-field *
8. General State Space*

6. Ergodic Theorems

1. Definitions and Examples
2. Birkhoff's Ergodic Theorem
3. Recurrence
4. A Subadditive Ergodic Theorem
5. Applications

7. Brownian Motion

1. Definition and Construction
2. Markov Property, Blumenthal's 0-1 Law
3. Stopping Times, Strong Markov Property
4. Maxima and Zeros
5. Martingales
6. Ito's formula*

8. Brownian Embeddings and Applications

1. Donsker's Theorem
2. CLTs for Martingales
3. CLTs for Stationary Sequences
4. Empirical Distributions, Brownian Bridge
5. Laws of the Iterated Logarithm

9. Multidimensional Brownian Motion

1. Martingales
2. Heat Equation
3. Inhomogenous Heat Equation
4. Feynman-Kac Fromula
5. Dirichlet Problem
6. Green's Functions and Potential Kernels
7. Poisson's Equation
8. Schrodinger Equation

Appendix: Measure Theory

1. Caratheodary's Extension Theorem
2. Which sets are measurable?
3. Kolmogorov's Extension Theorem
4. Radon-Nikodym Theorem
5. Differentiating Under the Integral