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A Basic Course in Probability Theory

  • Textbook
  • © 2016

Overview

Authors:
  1. Rabi Bhattacharya

    1. Department of Mathematics, University of Arizona, Tucson, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Edward C. Waymire

    1. Department of Mathematics, Oregon State Univeristy Department of Mathematics, Corvallis, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Quicker paced introduction to the basics allows for a more in-depth treatment of such topics as convergence theory and Brownian motion
  • Self-contained and suitable for students with varying levels of background in analysis and measure theory
  • Includes a complete overview of basic measure theory and analysis (with proofs), and an extensive bibliography for further reading in the area
  • Written in a lively and engaging style
  • Second edition has additional exercises and expanded basic theory, and a new chapter on general Markov dependent sequences
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xii
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  2. Random Maps, Distribution, and Mathematical Expectation

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 1-23

  3. Independence, Conditional Expectation

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 25-52

  4. Martingales and Stopping Times

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 53-74

  5. Classical Central Limit Theorems

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 75-85

  6. Classical Zero–One Laws, Laws of Large Numbers and Large Deviations

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 87-102

  7. Fourier Series, Fourier Transform, and Characteristic Functions

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 103-134

  8. Weak Convergence of Probability Measures on Metric Spaces

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 135-157

  9. Random Series of Independent Summands

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 159-166

  10. Kolmogorov’s Extension Theorem and Brownian Motion

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 167-178

  11. Brownian Motion: The LIL and Some Fine-Scale Properties

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 179-186

  12. Strong Markov Property, Skorokhod Embedding, and Donsker’s Invariance Principle

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 187-205

  13. A Historical Note on Brownian Motion

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 207-210

  14. Some Elements of the Theory of Markov Processes and Their Convergence to Equilibrium

    • Rabi Bhattacharya, Edward C. Waymire
    Pages 211-223

  15. Back Matter

    Pages 225-265
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Keywords

About this book

This text develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications. In this second edition, the text has been reorganized for didactic purposes, new exercises have been added and basic theory has been expanded.  General Markov dependent sequences and their convergence to equilibrium is the subject of an entirely new chapter. The introduction of conditional expectation and conditional probability very early in the text maintains the pedagogic innovation of the first edition; conditional expectation is illustrated in detail in the context of an expanded treatment of martingales, the Markov property, and the strong Markov property. Weak convergence of probabilities on metric spaces and Brownian motion are two topics to highlight. A selection of large deviation and/or concentration inequalities ranging from those of  Chebyshev, Cramer–Chernoff, Bahadur–Rao, to Hoeffding have been added,with illustrative comparisons of their use in practice. This also includes a treatment of the Berry–Esseen error estimate in the central limit theorem.


The authors assume mathematical maturity at a graduate level; otherwise the book is suitable for students with varying levels of background in analysis and measure theory. For the reader who needs refreshers, 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 a series of four upcoming graduate textbooks in stochastic processes with applications.

Authors and Affiliations

  • Department of Mathematics, University of Arizona, Tucson, USA

    Rabi Bhattacharya

  • Department of Mathematics, Oregon State Univeristy Department of Mathematics, Corvallis, USA

    Edward C. Waymire

About the authors

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.

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