A Basic Course in Probability Theory

by Rabi Bhattacharya, Edward C. Waymire

Overview

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.

About the Author:
Rabi Bhattacharya is Professor of Mathematics at the University of Arizona

About the Author:
Edward C. Waymire is Professor of Mathematics at Oregon State University

Product Details

ISBN-13:	9783319479743
Publisher:	Springer-Verlag New York, LLC
Publication date:	02/13/2017
Series:	Universitext
Sold by:	Barnes & Noble
Format:	eBook
File size:	14 MB
Note:	This product may take a few minutes to download.

About the Author

Rabi Bhattacharya, PhD, has held regular faculty positions at UC Berkeley; Indiana University; and the University of Arizona. He is a Fellow of the Institute of Mathematical Statistics and a recipient of the U.S. Senior Scientist Humboldt Award and of a Guggenheim Fellowship. He has served on editorial boards of many international journals and has published several research monographs and graduate texts on probability and statistics.

Edward C. Waymire, PhD, is Professor of Mathematics at Oregon State University. He received a PhD in mathematics from the University of Arizona in the theory of interacting particle systems. His primary research concerns applications of probability and stochastic processes to problems of contemporary applied mathematics pertaining to various types of flows, dispersion, and random disorder.
Both authors have co-authored numerous books, including a series of four upcoming graduate textbooks in stochastic processes with applications.

Table of Contents

Preface     vii
Random Maps, Distribution, and Mathematical Expectation     1
Exercises     14
Independence, Conditional Expectation     19
Exercises     33
Martingales and Stopping Times     37
Exercises     47
Classical Zero-One Laws, Laws of Large Numbers and Deviations     49
Exercises     57
Weak Convergence of Probability Measures     59
Exercises     70
Fourier Series, Fourier Transform, and Characteristic Functions     73
Exercises     93
Classical Central Limit Theorems     99
Exercises     104
Laplace Transforms and Tauberian Theorem     107
Exercises     118
Random Series of Independent Summands     121
Exercises     127
Kolmogorov's Extension Theorem and Brownian Motion     129
Exercises     138
Brownian Motion: The LIL and Some Fine-Scale Properties     141
Exercises     145
Skorokhod Embedding and Donsker's Invariance Principle     147
Exercises     164
A Historical Note on Brownian Motion     167
Measure and Integration     171
Measures and the Caratheodory Extension     171
Integration and Basic Convergence Theorems     176
Product Measures     182
Riesz Representation on C(S)     183
Topology and Function Spaces     187
Hilbert Spaces and Applications in Measure Theory     193
Hilbert spaces     193
Lebesgue Decomposition and the Radon-Nikodym Theorem     196
References     201
Index     205
Symbol Index     211

Editorial Reviews

From the reviews:

"Bhattacharya (Univ. of Arizona, Tucson) and Waymire (Oregon State Univ., Corvallis) write to provide the necessary probability background for studying stochastic processes. For students exposed to analysis and measure theory, the book can be used as a graduate-level course resource on probability. … Every chapter ends with a set of exercises, including numerous solved examples. Appendixes explain measure theory and integration, function spaces and topology, and Hilbert spaces and applications to measure theory. List of symbols. Summing Up: Recommended. Graduate students; faculty and researchers." (D. V. Chopra, CHOICE, Vol. 45 (7), 2008)

"The mentioned prerequisites are exposure to measure theory and analysis. Three appendices (29 pages) provide a brief but thorough introduction to the measure theory and functional analysis that is needed. … This well-written book is full of wonderful probability theory." (Kenneth A. Ross, MathDL, February, 2008)

"The book provides the fundamentals of probability theory in a measure-theoretic framework ... . is suitable for advanced undergraduate students and graduate students. The material is presented in a very dense and concise way and each chapter includes a section with exercises at the end ... . Thus the book may be used very well as a reference text and companion literature for a lecture course ... ." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1138 (16), 2008)

"This book is a self-contained exposition of various basic elements of probability theory. It is suitable for graduate students with some background in analysis, but may also serve as a quick reference for more experienced readers. … Overall, this book is quite rich and very pleasant to read … ." (Djalil Chafaï, Mathematical Reviews, Issue 2009 e)

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