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.

• Bhattacharya
• Brownian Motion
• Monty Hall problem
• Probability
• Probability theory
• Variance  Waymire
• measure theory

• Front Matter
• 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
• Back Matter

University of Arizona, Tucson, USA

Rabi Bhattacharya

Oregon State University, Corvallis, USA

Edward C. Waymire

Book Title：A Basic Course in Probability Theory

Authors：Rabi Bhattacharya , Edward C. Waymire

DOI：https://doi.org/10.1007/978-0-387-71939-9

eBook Packages：/

Edition Number：1

Number of Pages：XII, 220

Number of Illustrations：/

Topics:Probability Theory and Stochastic Processes, Measure and Integration, Analysis