Ebook: Probability theory: an advanced course

This book presents a selection of topics from probability theory. Essentially, the topics chosen are those that are likely to be the most useful to someone planning to pursue research in the modern theory of stochastic processes. The prospective reader is assumed to have good mathematical maturity. In particular, he should have prior exposure to basic probability theory at the level of, say, K.L. Chung's 'Elementary probability theory with stochastic processes' (Springer-Verlag, 1974) and real and functional analysis at the level of Royden's 'Real analysis' (Macmillan, 1968). The first chapter is a rapid overview of the basics. Each subsequent chapter deals with a separate topic in detail. There is clearly some selection involved and therefore many omissions, but that cannot be helped in a book of this size. The style is deliberately terse to enforce active learning. Thus several tidbits of deduction are left to the reader as labelled exercises in the main text of each chapter. In addition, there are supplementary exercises at the end. In the preface to his classic text on probability ('Probability', Addison­ Wesley, 1968), Leo Breiman speaks of the right and left hands of probability.;1 Introduction -- 1.1 Random Variables -- 1.2 Monotone Class Theorems -- 1.3 Expectations and Uniform Integrability -- 1.4 Independence -- 1.5 Convergence Concepts -- 1.6 Additional Exercises -- 2 Spaces of Probability Measures -- 2.1 The Prohorov Topology -- 2.2 Skorohod's Theorem -- 2.3 Compactness in P(S) -- 2.4 Complete Metrics on P(S) -- 2.5 Characteristic Functions -- 2.6 Additional Exercises -- 3 Conditioning and Martingales -- 3.1 Conditional Expectations -- 3.2 Martingales -- 3.3 Convergence Theorems -- 3.4 Martingale Inequalities -- 3.5 Additional Exercises -- 4 Basic Limit Theorems -- 4.1 Introduction -- 4.2 Strong Law of Large Numbers -- 4.3 Central Limit Theorem -- 4.4 The Law of Iterated Logarithms -- 4.5 Large Deviations -- 4.6 Tests for Convergence -- 4.7 Additional Exercises -- 5 Markov Chains -- 5.1 Construction and the Strong Markov Property -- 5.2 Classification of States -- 5.3 Stationary Distributions -- 5.4 Transient and Null Recurrent Chains -- 5.5 Additional Exercises -- 6 Foundations of Continuous-Time Processes -- 6.1 Introduction -- 6.2 Separability and Measurability -- 6.3 Continuous Versions -- 6.4 Cadlag Versions -- 6.5 Examples of Stochastic Processes -- 6.6 Additional Exercises -- References.