This popular textbook, now in a revised and expanded third edition, presents a comprehensive course in modern probability theory.

Probability plays an increasingly important role not only in mathematics, but also in physics, biology, finance and computer science, helping to understand phenomena such as magnetism, genetic diversity and market volatility, and also to construct efficient algorithms. Starting with the very basics, this textbook covers a wide variety of topics in probability, including many not usually found in introductory books, such as:

• limit theorems for sums of random variables
• martingales
• percolation
• Markov chains and electrical networks
• construction of stochastic processes
• Poisson point process and infinite divisibility
• large deviation principles and statistical physics
• Brownian motion
• stochastic integrals and stochastic differential equations.

The presentation is self-contained and mathematically rigorous, with the material on probability theory interspersed with chapters on measure theory to better illustrate the power of abstract concepts.This third edition has been carefully extended and includes new features, such as concise summaries at the end of each section and additional questions to encourage self-reflection, as well as updates to the figures and computer simulations. With a wealth of examples and more than 290 exercises, as well as biographical details of key mathematicians, it will be of use to students and researchers in mathematics, statistics, physics, computer science, economics and biology.

Table of Contents

1 Basic Measure Theory

2 Independence

3 Generating Functions

4 The Integral

5 Moments and Laws of Large Numbers

6 Convergence Theorems

7 Lp-Spaces and the Radon- Nikodym Theorem

8 Conditional Expectations

9 Martingales

10 Optional Sampling Theorems

11 Martingale Convergence Theorems and Their Applications

12 Backwards Martingales and Exchangeability

13 Convergence of Measures

14 Probability Measures on Product Spaces

15 Characteristic Functions and the Central Limit Theorem

16 Infinitely Divisible Distributions

17 Markov Chains

18 Convergence of Markov Chains

19 Markov Chains and Electrical Networks

20 Ergodic Theory

21 Brownian Motion

22 Law of the Iterated Logarithm

23 Large Deviations

24 The Poisson Point Process

25 The Ito Integral

26 Stochastic Differential Equations

About the Author

Achim Klenke is a professor at the Johannes Gutenberg University in Mainz, Germany. He is known for his work on interacting particle systems, stochastic analysis, and branching processes, in particular for his pioneering work with Leonid Mytnik on infinite rate mutually catalytic branching processes.