Probability Theory

Probability Theory : A Comprehensive Course

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Description

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.
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Product details

  • Paperback | 638 pages
  • 155 x 235 x 38.1mm | 929.86g
  • England, United Kingdom
  • English
  • Revised
  • 2nd ed. 2014
  • 1 Tables, black and white; 20 Illustrations, color; 26 Illustrations, black and white; XII, 638 p. 46 illus., 20 illus. in color.
  • 144715360X
  • 9781447153603
  • 483,968

Back cover copy

This second edition of the popular textbook contains a comprehensive course in modern probability theory. Overall, probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us in understanding magnetism, amorphous media, genetic diversity and the perils of random developments at financial markets, and they guide us in constructing more efficient algorithms. To address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, 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 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.
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Table of contents

Basic Measure Theory.- 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 Ito Integral.- Stochastic Differential Equations.
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Review Text

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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Review quote

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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About Achim Klenke

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

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