This is a textbook for a one-semester graduate course in measure-theoretic probability theory, but with ample material to cover an ordinary year-long course at a more leisurely pace. Khoshnevisan’s approach is to develop the ideas that are absolutely central to modern probability theory, and to showcase them by presenting their various applications. As a result, a few of the familiar topics are replaced by interesting non-standard ones. The topics range from undergraduate probability and classical limit theorems to Brownian motion and elements of stochastic calculus. Throughout, the reader will find many exciting applications of probability theory and probabilistic reasoning. There are numerous exercises, ranging from the routine to the very difficult. Each chapter concludes with historical notes.
Davar Khoshnevisan, University of Utah, Salt Lake City, UT
Preface General Notation Classical Probability Discrete Probability Conditional Probability Independence Discrete Distributions Absolutely Continuous Distributions Expectation and Variance Problems Notes Bernoulli Trials The Classical Theorems Problems Notes Measure Theory Measure Spaces LebesgueMeasure Completion Proof of Carath´eodory’s Theorem Problems Notes Integration Measurable Functions The Abstract Integral Lp-Spaces Modes of Convergence Limit Theorems The Radon–Nikod´ym Theorem Problems Notes Product Spaces Finite Products Infinite Products Complement: Proof of Kolmogorov’s Extension Theorem Problems Notes Independence Random Variables and Distributions Independent Random Variables An Instructive Example Khintchine’s Weak Law of Large Numbers Kolmogorov’s Strong Law of Large Numbers Applications Problems Notes The Central Limit Theorem Weak Convergence Weak Convergence and Compact-Support Functions Harmonic Analysis in Dimension One The Plancherel Theorem The 1-D Central Limit Theorem Complements to the CLT Problems Notes Martingales Conditional Expectations Filtrations and Semi-Martingales Stopping Times and Optional Stopping Applications to RandomWalks Inequalities and Convergence Further Applications Problems Notes BrownianMotion Gaussian Processes Wiener’s Construction: Brownian Motion on [0 ,1) Nowhere-Differentiability The Brownian Filtration and Stopping Times The StrongMarkov Property The Reflection Principle Problems Notes Terminus: Stochastic Integration The Indefinite Itˆo Integral Continuous Martingales in L2(P) The Definite Itˆo Integral Quadratic Variation Itˆo’s Formula and Two Applications Problems Notes Appendix Hilbert Spaces Fourier Series Bibliography Index