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Macmillan Higher Education Palgrave Higher Education

Probability Theory

A Comprehensive Course

Edition 2nd Edition
ISBN 9781447153603
Publication Date September 2013
Formats Paperback Ebook 
Publisher Springer

This second edition of the popular textbook contains a comprehensive course in modern probability theory, covering a wide variety of topics which are not usually found in introductory textbooks, including:
 • 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 integral and stochastic differential equations.

The theory is developed rigorously and in a self-contained way, with the chapters on measure theory interlaced with the probabilistic chapters in order to display the power of the abstract concepts in probability theory. This second edition has been carefully extended and includes many new features. It contains updated figures (over 50), computer simulations and some difficult proofs have been made more accessible. A wealth of examples and more than 270 exercises as well as biographic details of key mathematicians support and enliven the presentation. It will be of use to students and researchers in mathematics and statistics in physics, computer science, economics and biology.

Achim Klenke is a professor at the Johannes Gutenberg University in Mainz, Germany. 

Basic Measure Theory
Generating Functions
The Integral
Moments and Laws of Large Numbers
Convergence Theorems
Lp-Spaces and the Radon–Nikodym Theorem
Conditional Expectations
Optional Sampling Theorems
Martingale Convergence Theorems and Their Applications
Backwards Martingales and Exchangeability
Convergence of Measures
Probability Measures on Product Spaces
Characteristic Functions and the Central Limit Theorem
Infinitely Divisible Distributions
Markov Chains
Convergence of Markov Chains
Markov Chains and Electrical Networks
Ergodic Theory
Brownian Motion
Law of the Iterated Logarithm
Large Deviations
The Poisson Point Process
The Itˆo Integral
Stochastic Differential Equations.


From the book reviews:“The book is dedicated to graduate students who start to learn probability theory as well as to those who need an excellent reference book. … All results are presented in a self-contained way and are rigorously proved. Each section of the 26 chapters ends with a number of exercises, overall more than 270. … Altogether it is a very valuable book for all students who specialize in probability theory or statistics.” (Mathias Trabs, zbMATH, Vol. 1295, 2014)“The book under review is a standard graduate textbook in this area of mathematics that collects various classical and modern topics in a friendly volume. … the book contains many exercises. It is a very good source for a course in probability theory for advanced undergraduates and first-year graduate students. … the book should be useful for a wide range of audiences, including students, instructors, and researchers from all branches of science who are dealing with random phenomena.” (Mehdi Hassani, MAA Reviews, May, 2014)
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