Probability theory: a comprehensive course

Prof. Dr. Achim Klenke (auth.)1848000472, 9781848000476

Probabilistic concepts play an increasingly important role in mathematics, physics, biology, financial engineering and computer science. They help us to understand magnetism, amorphous media, genetic diversity and the perils of random developments on the financial markets, and they guide us in constructing more efficient algorithms.

This text is a comprehensive course in modern probability theory and its measure-theoretical foundations. Aimed primarily at graduate students and researchers, the book 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 processes and infinite divisibility;

large deviation principles and statistical physics;

Brownian motion; and

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 the world of probability theory. In addition, plenty of figures, computer simulations, biographic details of key mathematicians, and a wealth of examples support and enliven the presentation.

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Table of contents :
Front Matter….Pages I-XII
Basic Measure Theory….Pages 1-48
Independence….Pages 49-75
Generating Functions….Pages 77-84
The Integral….Pages 85-99
Moments and Laws of Large Numbers….Pages 101-127
Convergence Theorems….Pages 129-142
L p -Spaces and the Radon-Nikodym Theorem….Pages 143-167
Conditional Expectations….Pages 169-187
Martingales….Pages 189-203
Optional Sampling Theorems….Pages 205-215
Martingale Convergence Theorems and Their Applications….Pages 217-229
Backwards Martingales and Exchangeability….Pages 231-243
Convergence of Measures….Pages 245-270
Probability Measures on Product Spaces….Pages 271-292
Characteristic Functions and the Central Limit Theorem….Pages 293-326
Infinitely Divisible Distributions….Pages 327-343
Markov Chains….Pages 345-378
Convergence of Markov Chains….Pages 379-402
Markov Chains and Electrical Networks….Pages 403-430
Ergodic Theory….Pages 431-445
Brownian Motion….Pages 447-494
Law of the Iterated Logarithm….Pages 495-504
Large Deviations….Pages 505-524
The Poisson Point Process….Pages 525-542
The Itô Integral….Pages 543-566
Stochastic Differential Equations….Pages 567-590
Back Matter….Pages 591-621