A Basic Course in Probability Theory

• Main Authors: Bhattacharya, Rabi (Author), Waymire, Edward C. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: New York, NY : Springer New York, 2007.
• Series: Universitext
• Subjects: Mathematics.; Mathematical analysis.; Analysis (Mathematics).; Measure theory.; Probabilities.; Probability Theory and Stochastic Processes.; Measure and Integration.; Analysis.

 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. 

 Random Maps, Distribution, and Mathematical Expectation -- Independence, Conditional Expectation -- Martingales and Stopping Times -- Classical Zero–One Laws, Laws of Large Numbers and Deviations -- Weak Convergence of Probability Measures -- Fourier Series, Fourier Transform, and Characteristic Functions -- Classical Central Limit Theorems -- Laplace Transforms and Tauberian Theorem -- Random Series of Independent Summands -- Kolmogorov's Extension Theorem and Brownian Motion -- Brownian Motion: The LIL and Some Fine-Scale Properties -- Skorokhod Embedding and Donsker's Invariance Principle -- A Historical Note on Brownian Motion.