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 It^o Integral.- Stochastic Differential Equations.
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.rn rnTo address these concepts, the title covers a wide variety of topics, many of which are not usually found in introductory textbooks, such as:rn rn. limit theorems for sums of random variablesrn. martingalesrn. percolationrn. Markov chains and electrical networksrn. construction of stochastic processesrn. Poisson point process and infinite divisibilityrn. large deviation principles and statistical physicsrn. Brownian motionrn. stochastic integral and stochastic differential equations.rnrnThe 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.
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