Convergence of Stochastic Processes.

Abstract

I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct Approximation.- 3. The Combinatorial Method.- 4. Classes of Sets with Polynomial Discrimination.- 5. Classes of Functions.- 6. Rates of Convergence.- Notes.- Problems.- III Convergence in Distribution in Euclidean Spaces.- 1. The Definition.- 2. The Continuous Mapping Theorem.- 3. Expectations of Smooth Functions.- 4. The Central Limit Theorem.- 5. Characteristic Functions.- 6. Quantile Transformations and Almost Sure Representations.- Notes.- Problems.- IV Convergence in Distribution in Metric Spaces.- 1. Measurability.- 2. The Continuous Mapping Theorem.- 3. Representation by Almost Surely Convergent Sequences.- 4. Coupling.- 5. Weakly Convergent Subsequences.- Notes.- Problems.- V The Uniform Metric on Spaces of Cadlag Functions.- 1. Approximation of Stochastic Processes.- 2. Empirical Processes.- 3. Existence of Brownian Bridge and Brownian Motion.- 4. Processes with Independent Increments.- 5. Infinite Time Scales.- 6. Functional of Brownian Motion and Brownian Bridge.- Notes.- Problems.- VI The Skorohod Metric on D(0, ?).- 1. Properties of the Metric.- 2. Convergence in Distribution.- Notes.- Problems.- VII Central Limit Theorems.- 1. Stochastic Equicontinuity.- 2. Chaining.- 3. Gaussian Processes.- 4. Random Covering Numbers.- 5. Empirical Central Limit Theorems.- 6. Restricted Chaining.- Notes.- Problems.- VIII Martingales.- 1. A Central Limit Theorem for Martingale-Difference Arrays.- 2. Continuous Time Martingales.- 3. Estimation from Censored Data.- Notes.- Problems.- Appendix A Stochastic-Order Symbols.- Appendix B Exponential Inequalities.- Notes.- Problems.- Appendix C Measurability.- Notes.- Problems.- References.- Author Index.