Table of Contents

1 Mathematical Foundations.- 1.1 Introduction.- 1.2 Sets and Set Operations.- 1.3 Limits of Sequences.- 1.4 Measurable Spaces, Algebras, and Sets.- 1.5 Measures and Probability Measures.- 1.5.1 Measures and Measurable Functions.- 1.6 Integration.- 1.6.1 Miscellaneous Convergence Results.- 1.7 Extensions to Abstract Spaces.- 1.8 Miscellaneous Concepts.- 2 Foundations of Probability.- 2.1 Discrete Models.- 2.2 General Probability Models.- 2.2.1 The Measurable Space (Rn, Bn, Rn).- 2.2.2 Specification of Probability Measures.- 2.2.3 Fubini’s Theorem and Miscellaneous Results.- 2.3 Random Variables.- 2.3.1 Generalities.- 2.3.2 Random Elements.- 2.3.3 Moments of Random Variables and Miscellaneous Inequalities.- 2.4 Conditional Probability.- 2.4.1 Conditional Probability in Discrete Models.- 2.4.2 Conditional Probability in Continuous Models.- 2.4.3 Independence.- 3 Convergence of Sequences I.- 3.1 Convergence a.c. and in Probability.- 3.1.1 Definitions and Preliminaries.- 3.1.2 Characterization of Convergence a.c. and Convergence in Probability.- 3.2 Laws of Large Numbers.- 3.3 Convergence in Distribution.- 3.4 Convergence in Mean of Order p.- 3.5 Relations among Convergence Modes.- 3.6 Uniform Integrability and Convergence.- 3.7 Criteria for the SLLN.- 3.7.1 Sequences of Independent Random Variables.- 3.7.2 Sequences of Uncorrelated Random Variables.- 4 Convergence of Sequences II.- 4.1 Introduction.- 4.2 Properties of Random Elements.- 4.3 Base and Separability.- 4.4 Distributional Aspects of R.E.- 4.4.1 Independence for Random Elements.- 4.4.2 Distributions of Random Elements.- 4.4.3 Moments of Random Elements.- 4.4.4 Uncorrelated Random Elements.- 4.5 Laws of Large Numbers for R.E.- 4.5.1 Preliminaries.- 4.5.2 WLLN and SLLN for R.E.- 4.6 Convergence in Probability for R.E.- 4.7 Weak Convergence.- 4.7.1 Preliminaries.- 4.7.2 Properties of Measures.- 4.7.3 Determining Classes.- 4.7.4 Weak Convergence in Product Space.- 4.8 Convergence in Distribution for R.E.- 4.8.1 Convergence of Transformed Sequences of R.E.- 4.9 Characteristic Functions.- 4.10 CLT for Independent Random Variables.- 4.10.1 Preliminaries.- 4.10.2 Characteristic Functions for Normal Variables.- 4.10.3 Convergence in Probability and Characteristic Functions.- 4.10.4 CLT for i.i.d. Random Variables.- 4.10.5 CLT and the Lindeberg Condition.- 5 Dependent Sequences.- 5.1 Preliminaries.- 5.2 Definition of Martingale Sequences.- 5.3 Basic Properties of Martingales.- 5.4 Square Integrable Sequences.- 5.5 Stopping Times.- 5.6 Upcrossings.- 5.7 Martingale Convergence.- 5.8 Convergence Sets.- 5.9 WLLN and SLLN for Martingales.- 5.10 Martingale CLT.- 5.11 Mixing and Stationary Sequences.- 5.11.1 Preliminaries and Definitions.- 5.11.2 Measure Preserving Transformations.- 5.12 Ergodic Theory.- 5.13 Convergence and Ergodicity.- 5.14 Stationary Sequences and Ergodicity.- 5.14.1 Preliminaries.- 5.14.2 Convergence and Strict Stationarity.- 5.14.3 Convergence and Covariance Stationarity.- 5.15 Miscellaneous Results and Examples.