Topics in advanced econometrics : probability foundations

목차 (Table of Contents)

• CONTENTS
• Preface = Ⅴ
• 1 Mathematical Foundations = 1
• 1.1 Introduction = 1
• 1.2 Sets and Set Operations = 2
• 1.3 Limits of Sequences = 4
• 1.4 Measurable Spaces, Algebras, and Sets = .10
• 1.5 Measures and Probability Measures = 18
• 1.5.1 Measures and Measurable Functions = 18
• 1.6 Integration = 26
• 1.6.1 Miscellaneous Convergence Results = 43
• 1.7 Extensions to Abstract Spaces = 52
• 1.8 Miscellaneous Concepts = 64
• 2 Foundations of Probability = 74
• 2.1 Discrete Models = 74
• 2.2 General Probability Models = 77
• 2.2.1 The Measurable Space (R$$^n$$, B$$^n$$,R$$^n$$) = 77
• 2.2.2 Specification of Probability Measures = 82
• 2.2.3 Fubini's Theorem and Miscellaneous Results = 92
• 2.3 Random Variables = 97
• 2.3.1 Generalities = 97
• 2.3.2 Random Elements = 101
• 2.3.3 Moments of Random Variables and Miscellaneous Inequalities = 104
• 2.4 Conditional Probability = 110
• 2.4.1 Conditional Probability in Discrete Models = 110
• 2.4.2 Conditional Probability in Continuous Models = 120
• 2.4.3 Independence = 125
• 3 Convergence of Sequences Ⅰ = 133
• 3.1 Convergence a. c. and in Probability = 133
• 3.1.1 Definitions and Preliminaries = 133
• 3.1.2 Characterization of Convergence a. c. and Convergence in Probability = 140
• 3.2 Laws of Large Numbers = 152
• 3.3 Convergence in Distribution = 154
• 3.4 Convergence in Mean of Order p = 156
• 3.5 Relations among Convergence Modes = 160
• 3.6 Uniform Integrability and Convergence = 166
• 3.7 Criteria for the SLLN = 177
• 3.7.1 Sequences of Independent Random Variables = 177
• 3.7.2 Sequences of Un-correlated Random Variables = 191
• 4 Convergence of Sequences Ⅱ = 94
• 4.1 Introduction = 194
• 4.2 Properties of Random Elements = 196
• 4.3 Base and Separability = 200
• 4.4 Distributional Aspects of R.E. = 208
• 4.4.1 Independence for Random Elements = 208
• 4.4.2 Distributions of Random Elements = 210
• 4.4.3 Moments of Random Elements = 212
• 4.4.4 Un-correlated Random Elements = 216
• 4.5 Laws of Large Numbers for R. E. = 218
• 4.5.1 Preliminaries = 218
• 4.5.2 WLLN and SLLN for R. E. = 219
• 4.6 Convergence in Probability for R. E. = 222
• 4.7 Weak Convergence = 223
• 4.7.1 Preliminaries = 223
• 4.7.2 Properties of Measures = 226
• 4.7.3 Determining Classes = 231
• 4.7.4 Weak Convergence in Product Space = 234
• 4.8 Convergence in Distribution for R. E. = 236
• 4.8.1 Convergence of Transformed Sequences of R. E. = 239
• 4.9 Characteristic Functions = 246
• 4.10 CLT for Independent Random Variables = 257
• 4.10.1 Preliminaries = 257
• 4.10.2 Characteristic Functions for Normal Variables = 259
• 4.10.3 Convergence in Probability and Characteristic Functions = 261
• 4.10.4 CLT for i.i.d. Random Variables = 263
• 4.10.5 CLT and the Lindeberg Condition = 265
• 5 Dependent Sequences = 277
• 5.1 Preliminaries = 277
• 5.2 Definition of Martingale Sequences = 279
• 5.3 Basic Properties of Martingales = 282
• 5.4 Square Integrable Sequences = 285
• 5.5 Stopping Times = 289
• 5.6 Upcrossings = 304
• 5.7 Martingale Convergence = 306
• 5.8 Convergence Sets = 310
• 5.9 WLLN and SLLN for Martingales = 320
• 5.10 Martingale CLT = 323
• 5.11 Mixing and Stationary Sequences = 338
• 5.11.1 Preliminaries and Definitions = 338
• 5.11.2 Measure Preserving Transformations = 342
• 5.12 Ergodic Theory = 344
• 5.13 Convergence and Ergodicity = 349
• 5.14 Stationary Sequences and Ergodicity = 355
• 5.14.1 Preliminaries = 355
• 5.14.2 Convergence and Strict Stationarity = 357
• 5.14.3 Convergence and Covariance Stationarity = 359
• 5.15 Miscellaneous Results and Examples = 366
• Bibliography = 371
• Index = 373