Topics in Advanced Econometrics : Probability Foundations

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 -- 1.6 Integration -- 1.7 Extensions to Abstract Spaces -- 1.8 Miscellaneous Concepts -- 2 Foundations of Probability -- 2.1 Discrete Models -- 2.2 General Probability Models -- 2.3 Random Variables -- 2.4 Conditional Probability -- 3 Convergence of Sequences I -- 3.1 Convergence a.c. and 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 -- 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 -- 4.5 Laws of Large Numbers for R.E -- 4.6 Convergence in Probability for R.E -- 4.7 Weak Convergence -- 4.8 Convergence in Distribution for R.E -- 4.9 Characteristic Functions -- 4.10 CLT for Independent Random Variables -- 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 -- 5.12 Ergodic Theory -- 5.13 Convergence and Ergodicity -- 5.14 Stationary Sequences and Ergodicity -- 5.15 Miscellaneous Results and Examples.