A Basic Course in Probability Theory

✍ Scribed by Rabi Bhattacharya, Edward C. Waymire,

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 223
• Series: Universitext
• Edition: 1
• Category: Library

✦ Synopsis

Introductory Probability is a pleasure to read and provides a fine answer to the question: How do you construct Brownian motion from scratch, given that you are a competent analyst? There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path.

✦ Table of Contents

Cover......Page 1
A Basic Course
in Probability Theory......Page 4
ISBN 0387719385......Page 5
PREFACE......Page 7
Contents......Page 10
1 Random Maps, Distribution, and Mathematical Expectation......Page 12
2 Independence, Conditional Expectation......Page 30
3 Martingales and Stopping Times......Page 48
4 Classical Zero–One Laws, Laws of Large Numbers and Large Deviations......Page 60
5 Weak Convergence of Probability Measures......Page 70
6 Fourier Series, Fourier Transform, and Characteristic Functions......Page 84
7 Classical Central Limit Theorems......Page 110
8 Laplace Transforms and Tauberian Theorem......Page 118
9 Random Series of Independent Summands......Page 132
10 Kolmogorov’s Extension Theorem and Brownian Motion......Page 140
11 Brownian Motion: The LIL and Some Fine-Scale Properties......Page 152
12 Skorokhod Embedding and Donsker’s Invariance Principle......Page 158
13 A Historical Note on Brownian Motion......Page 178
Appendix A: Measure and Integration......Page 182
Appendix B: Topology and Function Spaces......Page 198
Appendix C: Hilbert Spaces and Applications in Measure Theory......Page 204
References......Page 212
Index......Page 216
Symbol Index......Page 222