An Introduction to Measure-Theoretic Probability

Description

An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching the basics of measure theoretic probability. This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas should be equipped with. This edition requires no prior knowledge of measure theory, covers all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation. Topics range from the basic properties of a measure to modes of convergence of a sequence of random variables and their relationships; the integral of a random variable and its basic properties; standard convergence theorems; standard moment and probability inequalities; the Hahn-Jordan Decomposition Theorem; the Lebesgue Decomposition T; conditional expectation and conditional probability; theory of characteristic functions; sequences of independent random variables; and ergodic theory. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. Extensive exercises and practical examples are included, and all proofs are presented in full detail. Complete and detailed solutions to all exercises are available to the instructors on the book companion site. This text will be a valuable resource for graduate students primarily in statistics, mathematics, electrical and computer engineering or other information sciences, as well as for those in mathematical economics/finance in the departments of economics. Front Cover Half title Title Page Copyright Dedication Contents Pictured on the Cover Preface to First Edition Preface to Second Edition 1 Certain Classes of Sets, Measurability, and Pointwise Approximation 1.1 Measurable Spaces 1.2 Product Measurable Spaces 1.3 Measurable Functions and Random Variables 2 Definition and Construction of a Measure and its Basic Properties 2.1 About Measures in General, and Probability Measures in Particular 2.2 Outer Measures 2.3 The Carathéodory Extension Theorem 2.4 Measures and (Point) Functions 3 Some Modes of Convergence of Sequences of Random Variables and their Relationships 3.1 Almost Everywhere Convergence and Convergence in Measure 3.2 Convergence in Measure is Equivalent to Mutual Convergence 4 The Integral of a Random Variable and its Basic Properties 4.1 Definition of the Integral 4.2 Basic Properties of the Integral 4.3 Probability Distributions 5 Standard Convergence Theorems, The Fubini Theorem 5.1 Standard Convergence Theorems and Some of Their Ramifications 5.2 Sections, Product Measure Theorem, the Fubini Theorem 5.2.1 Preliminaries for the Fubini Theorem 6 Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications 6.1 Moment and Probability Inequalities 6.2 Convergence in the rth Mean, Uniform Continuity 7 The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem 7.1 The Hahn–Jordan Decomposition Theorem 7.2 The Lebesgue Decomposition Theorem 7.3 The Radon–Nikodym Theorem 8 Distribution Functions and Their Basic Properties, Helly–Bray Type Results 8.1 Basic Properties of Distribution Functions 8.2 Weak Convergence and Compactness 8.3 Helly–Bray Type Theorems for Distribution Functions 9 Conditional Expectation and Conditional Probability, and Related Properties and Results 9.1 Definition of Conditional Expectation and Conditional Probability 9.2 Some Basic Theorems About Conditional Expectations 9.3 Convergence Theorems and Inequalities for Conditional Expectations 9.4 Further Properties of Conditional Expectations 10 Independence 10.1 Independence of Events, σ-Fields, and Random Variables 10.2 Some Auxiliary Results 10.3 Proof of Theorem 1 and of Lemma 1 in Chapter 9 11 Topics from the Theory of Characteristic Functions 11.1 Definition of the Characteristic Function of a Distribution and Basic Properties 11.2 The Inversion Formula 11.3 Convergence in Distribution and Convergence 11.4 Convergence in Distribution in the Multidimensional Case 11.5 Convolution of Distribution Functions and Related Results 11.6 Some Further Properties of Characteristic Functions 11.7 Applications to the Weak Law of Large Numbers 11.8 The Moments of a Random Variable Determine its Distribution 11.9 Some Basic Concepts and Results from Complex Analysis 12 The Central Limit Problem: The Centered Case 12.1 Convergence to the Normal Law (Central Limit Theorem, CLT) 12.2 Limiting Laws of Under Conditions 12.3 Conditions for the Central Limit Theorem to Hold 12.4 Proof of Results in Section 12.2 13 The Central Limit Problem: The Noncentered Case 13.1 Notation and Preliminary Discussion 13.2 Limiting Laws of Under Conditions () 13.3 Two Special Cases of the Limiting Laws of 14 Topics from Sequences of Independent Random Variables 14.1 Kolmogorov Inequalities 14.2 More Important Results Toward Proving the Strong 14.3 Statement and Proof of the Strong Law of Large Numbers 14.4 A Version of the Strong Law of Large Numbers 14.5 Some Further Results on Sequences of Independent 15 Topics from Ergodic Theory 15.1 Stochastic Process, the Coordinate Process, Stationary Process 15.2 Measure-Preserving Transformations, the Shift Transformation 15.3 Invariant and Almost Sure Invariant Sets Relative to a Transformation 15.4 Measure-Preserving Ergodic Transformations, Invariant Random 15.5 The Ergodic Theorem, Preliminary Results 15.6 Invariant Sets and Random Variables Relative to a Process 16 Two Cases of Statistical Inference: Estimation of a Real-Valued Parameter, Nonparametric Estimation of a Probability Density Function 16.1 Construction of an Estimate of a Real-Valued Parameter 16.2 Construction of a Strongly Consistent Estimate 16.3 Some Preliminary Results 16.4 Asymptotic Normality of the Strongly Consistent Estimate 16.5 Nonparametric Estimation of a Probability Density Function 16.6 Proof of Theorems 3–5 Appendix A: Brief Review of Chapters 1–16 Appendix B: Brief Review of Riemann–Stieltjes Integral Appendix C: Notation and Abbreviations Selected References Revised Answers Manual to an Introduction to Measure-Theoretic Probability Index A B C D E F G H I J K L M N O P Q R S T U V W