Probability Theory: A Comprehensive Course (Universitext)

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. 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. Front Matter....Pages I-XII Basic Measure Theory....Pages 1-45 Independence....Pages 47-75 Generating Functions....Pages 77-84 The Integral....Pages 85-99 Moments and Laws of Large Numbers....Pages 101-130 Convergence Theorems....Pages 131-143 L p -Spaces and the Radon–Nikodym Theorem....Pages 145-168 Conditional Expectations....Pages 169-188 Martingales....Pages 189-203 Optional Sampling Theorems....Pages 205-215 Martingale Convergence Theorems and Their Applications....Pages 217-230 Backwards Martingales and Exchangeability....Pages 231-243 Convergence of Measures....Pages 245-271 Probability Measures on Product Spaces....Pages 273-293 Characteristic Functions and the Central Limit Theorem....Pages 295-330 Infinitely Divisible Distributions....Pages 331-349 Markov Chains....Pages 351-388 Convergence of Markov Chains....Pages 389-410 Markov Chains and Electrical Networks....Pages 411-438 Ergodic Theory....Pages 439-456 Brownian Motion....Pages 457-508 Law of the Iterated Logarithm....Pages 509-519 Large Deviations....Pages 521-541 The Poisson Point Process....Pages 543-561 The Itô Integral....Pages 563-588 Stochastic Differential Equations....Pages 589-611 Back Matter....Pages 613-638