## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

## Book Description

This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The new edition begins with a short chapter on measure theory to orient readers new to the subject.

1. Cover
2. Title Page
4. Contents
5. Preface
6. 1 Measure Theory
1. 1.1 Probability Spaces
2. 1.2 Distributions
3. 1.3 Random Variables
4. 1.4 Integration
5. 1.5 Properties of the Integral
6. 1.6 Expected Value
7. 1.7 Product Measures, Fubini’s Theorem
7. 2 Laws of Large Numbers
1. 2.1 Independence
2. 2.2 Weak Laws of Large Numbers
3. 2.3 Borel-Cantelli Lemmas
4. 2.4 Strong Law of Large Numbers
5. 2.5 Convergence of Random Series*
6. 2.6 Large Deviations*
8. 3 Central Limit Theorems
1. 3.1 The De Moivre-Laplace Theorem
2. 3.2 Weak Convergence
3. 3.3 Characteristic Functions
4. 3.4 Central Limit Theorems
5. 3.5 Local Limit Theorems*
6. 3.6 Poisson Convergence
7. 3.7 Stable Laws*
8. 3.8 Infinitely Divisible Distributions*
9. 3.9 Limit Theorems in R[sup(d)]
9. 4 Random Walks
10. 5 Martingales
1. 5.1 Conditional Expectation
2. 5.2 Martingales, Almost Sure Convergence
3. 5.3 Examples
4. 5.4 Doob’s Inequality, Convergence in L[sup(p)]
5. 5.5 Uniform Integrability, Convergence in L[sup(1)]
6. 5.6 Backwards Martingales
7. 5.7 Optional Stopping Theorems
11. 6 Markov Chains
1. 6.1 Definitions
2. 6.2 Examples
3. 6.3 Extensions of the Markov Property
4. 6.4 Recurrence and Transience
5. 6.5 Stationary Measures
6. 6.6 Asymptotic Behavior
7. 6.7 Periodicity, Tail σ-field*
8. 6.8 General State Space*
12. 7 Ergodic Theorems
13. 8 Brownian Motion
1. 8.1 Definition and Construction
2. 8.2 Markov Property, Blumenthal’s 0-1 Law
3. 8.3 Stopping Times, Strong Markov Property
4. 8.4 Path Properties
5. 8.5 Martingales
6. 8.6 Donsker’s Theorem
7. 8.7 Empirical Distributions, Brownian Bridge
8. 8.8 Laws of the Iterated Logarithm*
14. Appendix A: Measure Theory Details
15. References
16. Index