Probability: Theory and Examples. 5th Edition pdf 在作者主页可下载https://services.math.duke.edu/~rtd/PTE/pte5.html

Probability Theory and Examples 5th Edition.pdf

Durrett概率论第五版20175月版横空出世。美国杜克大学Rick Durrett教授所著此书反映了过去半个多世纪里概率论与随机过程的巨大发展，体现了概率论与其他学科深刻联系以及在工程、经济、金融等方面的应用，继承了美国在概率论教育实践中所积累的经验。内容包括大数定律、中心极限定理、随机游动、鞅论、马氏链、遍历定理、布朗运动、重对数律、与PDE的联系等。附录部分收录了所需的测度论知识。宜为概率统计专业研究生《高等概率论》和《随机过程论》的教材。对于学过概率论的学者而言， 这也不失为一本出色的参考书。

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* (was Section 4.4)
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 (was Subsection 3.6.3)
8. Stable Laws*
9. Infinitely Divisible Distributions*
10. Limit Theorems in Rd *

4. Martingales
1. Conditional Expectation
2. Martingales, Almost Sure Convergence
3. Examples
4. Doob's Inequality, Lp 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 (was Section 4.3)

5. Markov Chains
1. Examples (was Section 5.2)
2. Construction, Markov Properties (combines 5.1 and 5.3)
3. Recurrence and Transience
4. Recurrence of Random Walks (was Section 4.2)
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 (from Third Edition)
3. CLTs for Stationary Sequences (from Third Edition)
4. Empirical Distributions, Brownian Bridge*
5. Laws of the Iterated Logarithm

9. Multidimensional Brownian Motion (new chapter)
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


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