Klenke A. Probability Theory: A Comprehensive Course

Springer-Verlag London, 2008, 616 pages, ISBN: 1848000472Probabilistic 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.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