Mathematics of Probability

This book covers the basics of modern probability theory. It begins with probability theory on finite and countable sample spaces and then passes from there to a concise course on measure theory, which is followed by some initial applications to probability theory, including independence and conditional expectations. The second half of the book deals with Gaussian random variables, with Markov chains, with a few continuous parameter processes, including Brownian motion, and, finally, with martingales, both discrete and continuous parameter ones. The book is a self-contained introduction to probability theory and the measure theory required to study it.

Table of Contents

Chapter 1. Some Background and Preliminaries
Chapter 2. Probability Theory on Uncountable Sample Spaces
Chapter 3. Some Applications to Probability Theory
Chapter 4. The Central Limit Theorem and Gaussian Distributions
Chapter 5. Discrete Parameter Stochastic Processes
Chapter 6. Some Continuous-Time Processes
Chapter 7. Martingales