George G. Roussas University of California, Davis

Transcription

1 AN INTRODUCTION TO MEASURE-THEORETIC PROBABILITY George G. Roussas University of California, Davis

2 TABLE OF CONTENTS PREFACE xi CHAPTER I: Certain Classes of Sets, Measurability, and Pointwise Approximation 1 1. Measurable spaces 1 2. Product measurable spaces 8 3. Measurable functions and random variables 11 Exercises 20 CHAPTER 2: Definition and Construction of a Measure and Its Basic Properties About measures in general, and probability measures in particular Outer measures : The Caratheodory extension theorem Measures and (point) functions 43 Exercises 48 CHAPTER 3: Some Modes of Convergence of Sequences of Random Variables and Their Relationships Almost everywhere convergence and convergence in measure Convergence in measure is equivalent to mutual convergence in measure 60 Exercises 67 CHAPTER 4: The Integral of a Random Variable and Its Basic Properties Definition of the integral Basic properties of the integral Probability distributions 84 Exercises 86 VII

3 CHAPTER 5: Standard Convergence Theorems, the Fubini Theorem Standard convergence theorems and some of their ramifications Sections, Product Measure Theorem, the Fubini Theorem 102 Exercises 114 CHAPTER 6: Standard Moment and Probability Inequalities, Convergence in the rth Mean and Its Implications Moment and probability inequalities Convergence in the rth mean, uniform continuity, uniform integrability, and their relationships 127 Exercises 141 CHAPTER 7: The Hahn-Jordan Decomposition Theorem, the Lebesgue Decomposition Theorem, and the Radon Nikodym Theorem The Hahn Jordan Decomposition Theorem The Lebesgue Decomposition Theorem The Radon-Nikodym Theorem 161 Exercises 165 CHAPTER 8: Distribution Functions and Their Basic Properties, Helly-Bray Type Results Basic properties of distribution functions Weak convergence and compactness of a sequence of distribution functions Helly Bray type theorems for distribution functions 179 Exercises 184 CHAPTER 9: Conditional Expectation and Conditional Probability, and Related Properties and Results Definition of conditional expectation and conditional probabilitiy Some basic theorems about conditional expectations and conditional probabilities Convergence theorems and inequalities for conditional expectations Further properties of conditional expectations and conditional probabilities 204 Exercises 211

4 Table of Contents CHAPTER 10: Independence Independence of events, cr-fields, and random variables Some auxiliary results Proof of Theorem 1, and of Lemma 1 in Chapter Exercises : 229 CHAPTER 11: Topics from the Theory of Characteristic Functions Definition of the characteristic function of a distribution and basic properties The inversion formula Convergence in distribution and convergence of characteristic functions the Paul Levy continuity theorem, Convergence in distribution in the multidimentional case the Cramer Wold device Convolution of distribution functions and related results Some further properties of characteristic functions Applications to the Weak Law of Large Numbers and the Central Limit Theorem The moments of a random variable determine its distribution Some basic concepts'and results from complex analysis employed in the proof of Theorem Exercises 282 CHAPTER 12: The Central Limit Problem: The Centered Case Convergence to the normal law (Central Limit Theorem, CLT) Limiting laws of (S n ) under conditions (C) Conditions for the Central Limit Theorem to hold Proof of results in Section Exercises 321 CHAPTER 13: The Central Limit Problem: The Noncentered Case Notation and preliminary discussion Limiting laws of C(S n ) under conditions (C") Two special cases of the limiting laws of C(S n ) 334 Exercises 342

5 Table of Contents CHAPTER 14: Topics from Sequences of Independent Random Variables Kolmogorov inequalities More important results toward proving the Strong Law of Large Numbers Statement and proof of the Strong Law of Large Numbers A version of the Strong Law of Large Numbers for random variables with infinite expectation Some further results on sequences of independent random variables 374 Exercises 381 CHAPTER 15: Topics from Ergodic Theory Stochastic processes, the coordinate process, stationary process, and related results Measure-preserving transformations, the shift transformation, and related results Invariant and almost sure invariant sets relative to a transformation and related results Measure-preserving ergodic transformations, Invariant random variables relative to a transformation and related results The Ergodic Theorem, preliminary results Invariant sets and random variables relative to a process, formulation of the Ergodic Theorem in terms of a stationary process, ergodic processes 410 Exercises 418 Appendix 421 Selected References 431 Index 433