概率论

Introductory Lecture

1 Laws of Large Numbers

1. Basic Definitions

2. Random Variables

3. Expected Value

a. Inequalities

b. Integration to the limit

c. Computing expected values

4. Independence

a. Sufficient conditions for independence

b. Independence, distribution, and expectation

c. Constructing independent random variables

5. Weak Laws of Large Numbers

a. L2 weak laws

b. Triangular arrays

c. Truncation

6. Borel-Cantelli Lemmas

7. Strong Law of Large Numbers

8. Convergence of Random Series

9. Large Deviations

2 Central Limit Theorems

1. The De Moivre-Laplace Theorem

2. Weak Convergence

a. Examples

b. Theory

3. Characteristic Functions

a. Definition, inversion formula

b. Weak convergence

c. Moments and derivatives

d. Polyas criterion

e. The moment problem

4. Central Limit Theorems

a. i.i.d, sequences

b. Triangular arrays

c. Prime divisors (Erd0s-Kac)

d. Rates of convergence (Berry-Esseen)

5. Local Limit Theorems

6. Poisson Convergence

a. Basic limit theorem

b. Two examples with dependence

c. Poisson processes

7. Stable Laws

8. Infinitely Divisible Distributions

9. Limit theorems in ad

3 Random Walks

1. Stopping Times

2. Recurrence

3. Visits to 0, Arcsine Laws

4. Renewal Theory

4 Martingales

1. Conditional Expectation

a. Examples

b. Properties

c. Regular conditional probabilities

2. Martingales, Almost Sure Convergence

3. Examples 236

a. Bounded increments

b. Polyas urn scheme

c. Radon-Nikodym derivatives

d. Branching processes

4. Doobs Inequality, Lp Convergence Square integrable martingales

5. Uniform Integrability, Convergence in L1

6. Backwards Martingales

7. Optional Stopping Theorems

Markov Chains

Ergodic Theorems

Brownian Motion

Appendix:Measure Theory

References

Notation

Normal Table

Index

本书由美国康奈尔大学Rick Durrett教授撰写，书中反映了过去半个多世纪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。本书选材恰当，编排合理，难度适中，兼顾理论与应用，契合当今研究生教学的实际情况，被美国多所著名高校选为研究生教材。
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美国康奈尔大学Rick Durrett教授所著Probability： Theory and Examples一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。《概率论》(第3版)选材恰当，编排合理，难度适中，兼顾理论与应用，契合当今研究生教学的实际情况，为美国多所著名高校选为研究生教材。
注：《概率论》(第3版)为全英文版。