A Basic Course in Probability Theory (Universitext)

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. The book develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications. With this goal in mind, the pace is lively, yet thorough. Basic notions of independence and conditional expectation are introduced relatively early on in the text, while conditional expectation is illustrated in detail in the context of martingales, Markov property and strong Markov property. Weak convergence of probabilities on metric spaces and Brownian motion are two highlights. The historic role of size-biasing is emphasized in the contexts of large deviations and in developments of Tauberian Theory. The authors assume a graduate level of maturity in mathematics, but otherwise the book will be suitable for students with varying levels of background in analysis and measure theory. In particular, theorems from analysis and measure theory used in the main text are provided in comprehensive appendices, along with their proofs, for ease of reference. Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. Edward Waymire is Professor of Mathematics at Oregon State University. Both authors have co-authored numerous books, including the graduate textbook, Stochastic Processes with Applications 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