A Basic Course in Probability Theory

Description

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. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set. Cover......Page 1 Universitext......Page 3 A Basic Course in Probability Theory......Page 4 PREFACE......Page 7 Contents......Page 10 1 Random Maps, Distribution, and Mathematical Expectation......Page 12 2 Independence, Conditional Expectation......Page 29 3 Martingales and Stopping Times......Page 47 4 Classical Zero–One Laws, Laws of Large Numbers and Deviations......Page 59 5 Weak Convergence of Probability Measures......Page 69 6 Fourier Series, Fourier Transform, and Characteristic Functions......Page 83 7 Classical Central Limit Theorems......Page 109 8 Laplace Transforms and Tauberian Theorem......Page 116 9 Random Series of Independent Summands......Page 130 10 Kolmogorov's Extension Theorem and Brownian Motion......Page 137 11 Brownian Motion The LIL and Some Fine-Scale Properties......Page 149 12 Skorokhod Embedding and Donsker's Invariance Principle......Page 155 13 A Historical Note on Brownian Motion......Page 174 A P P E N D I X A Measure and Integration......Page 178 A P P E N D I X B Topology and Function Spaces......Page 194 A P P E N D I X C Hilbert Spaces and Applicationsin Measure Theory......Page 199 References......Page 206 Index......Page 210 Symbol Index......Page 216 Universitext......Page 217