## Probability Theory (2nd Edition)

#### A Comprehensive Course

**Author(s):**

Achim Klenke

**Publisher:**

Springer

**Pages:**638

**Further Actions:**

**Categories:**

### AVAILABLE FORMATS

Paperback - 9781447153603

30 August 2013

* €74.99*

Free Shipping

**In stock**

Ebook - 9781447153610

30 August 2013

* €63.06*

**In stock**

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...

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.

Show Less

Presents an updated, comprehensive and modern introduction to the most important fields of probability theory

Contains many new figures and examples

Studies a wide variety of topics on probability theory, many of which are not found in introductory textbooks

Independence

Generating Functions

The Integral

Moments and Laws of Large Numbers

Convergence Theorems

Lp-Spaces and the Radon–Nikodym Theorem

Conditional Expectations

Martingales

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)