Probability Theory: An Advanced Course (Paperback, Softcover Repri)

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.