【目 录】
Pictured on the Cover
Preface to First Edition
Preface to Second Edition
CHAPTER 1 Certain Classes of Sets, Measurability, and Pointwise Approximation //1
1.1 Measurable Spaces //1
1.2 Product Measurable Spaces //5
1.3 Measurable Functions and Random Variables //7
CHAPTER 2 Definition and Construction of a Measure and its Basic Properties //19
2.1 About Measures in General, and Probability Measures in Particular //19
2.2 Outer Measures //22
2.3 The Carathéodory Extension Theorem //27
2.4 Measures and (Point) Functions //30
CHAPTER 3 Some Modes of Convergence of Sequences of Random Variables and their Relationships //41
3.1 Almost Everywhere Convergence and Convergence in Measure //41
3.2 Convergence in Measure is Equivalent to Mutual Convergence in Measure //45
CHAPTER 4 The Integral of a Random Variable and its Basic Properties //55
4.1 Definition of the Integral //55
4.2 Basic Properties of the Integral //60
4.3 Probability Distributions //66
CHAPTER 5 Standard Convergence Theorems，The Fubini Theorem //71
5.1 Standard Convergence Theorems and Some of Their Ramifications //71
5.2 Sections, Product Measure Theorem，the Fubini Theorem //80
5.2.1 Preliminaries for the Fubini Theorem //88
CHAPTER 6 Standard Moment and Probability inequalities，Convergence in the rth Mean and its Implications //95
6.1 Moment and Probability Inequalities //95
6.2 Convergence in the rth Mean，Uniform Continuity，Uniform Integrability，and Their Relationships //101
CHAPTER 7 The Hahn-Jordan Decomposition Theorem，The Lebesgue Decomposition Theorem，and the Radon-Nikodym Theorem //117
7.1 The Hahn-JordanDecomposition Theorem //117
7.2 The Lebesgue Decomposition Theorem //122
7.3 The Radon-Nikodym Theorem //128
CHAPTER 8 Distribution Functions and Their Basic Properties，Helly-Bray Type Results //135
8.1 Basic Properties of Distribution Functions //135
8.2 Weak Convergence and Compactness of a Sequence of Distribution Functions //141
8.3 Helly-Bray Type Theorems for Distribution Functions //145
CHAPTER 9 Conditional Expectation and Conditional Probability，and Related Properties and Results //153
9.1 Definition of Conditional Expectation and Conditional Probability //153
9.2 Some Basic Theorems About Conditional Expectations and Conditional Probabilities //158
9.3 Convergence Theorems and Inequalities for Conditional Expectations //160
9.4 Furth Properties of Conditional Expectations and Conditional Probabilities //169
CHAPTER 10Independence//179
10.1 Independence of Events，σ-Fields，and Random Variables //179
10.2 Some Auxiliary Results //181
10.3hoof of Theorem 1 and of Lemma1 inChapter 9 //187
CHAPTER 11 Topics from the Theory of Characteristic Functions //193
11.1 Definition of the Characteristic Function of a Distribution and Basic Properties //193
11.2 The Inversion Formula //195
11.3 Convergence in Distribution and Convergence of Characteristic Functions-The Paul Lévy Continuity Theorem//202
11.4 Convergence in Distribution in the Multidimensional Case-The Cramér'-Wold Device //210
11.5 Convolution of Distribution Functions and Related Results //211
11.6 Some Further Properties of Characteristic Functions //216
11.7 Applications to the Weak Law of Large Numbers and the Central Lirnit Theorem //223
11.8 The Moments of a Random Variable Detenmne its Distribution //225
11.9 Some Basic Concepts and Results from Complex Analysis Employed in the Proof of Theorem 11 //229
CHAPTER 12 The Central limit Problem：The Centered Case //239
12.1 Convergence to the Normal Law (Central Limit Theorem，CLT) //240
12.2 Limiting Laws of L(Sn) Under Conditions (C) //245
12.3 Conditions for the Central Limit Theorem to Hold //252
12.4 Proof of Results in Section 12.2 //260
CHAPTER 13 The central Limit Problem：The Noncentered Case //271
13.1 Notation and Preliminary Discuss1on //271
13.2 Limiting Laws of L(Sn) Under COnditions (C") //274
13.3 Two Special Cases of the Limiting Laws of L(Sn) //278
CHAPTER 14 Topics from Sequences of Independent Random Variables //290
14.1 Kolmogorov Inequalities //290
14.2 More Important Results Toward Proving the Strong Law of Large Numbers //294
14.3 Statement and proof of the Strong Law of Large Numbers //302
14.4 A Version of the Strong Law of Large Numbers for Random Variables with Infinite Expectation //309
14.5 Some Further Results on Sequences of Independent Random Variables //313
CHAPTER 15 Topics from Ergodic Theory //319
15.1 Stochastic Process, the Coordinate Process，Stationary Process，and Related Results //320
15.2 Measure-Preserving Transformations，the Shift Transformation，and Related Results //323
15.3 Invariant and Almost Sure Invariant Sets Relative to a Transformation，and Related Results //326
15.4 Measure-Preserving Ergodic Transformations，Invariant Random Variables Relative to a Transformation, and Related Results //331
15.5 The Ergodic Theorem，Preliminary Results //332
15.6 Invariant Sets and Random Variables Relative to a Process，Formulation of the Ergodic Theorem in Terms of Stationary Processes，Ergodic Processes //340
CHAPTER 16 Two Cases of Statistical Inference：Estimation of a Real-Valued Parameter，Nonparametric Estimation of a Probability Density Function //347
16.1 Construction of an Estimate of a Real-Valued Parameter //347
16.2 Construction of a Strongly Consistent Estimate of a Real-Valued Parameter //348
16.3 Some Preliminary Results //351
16.4 Asymptotic Normality of the Strongly Consistent Estimate //355
16.5 Nonparametric Estimation of a Probability Density Function //364
16.6 Proof of Theorems 3-5 //368
APPENDIX A Brief Review of Chapters 1-16 //375
APPENDIX B Brief Review of Riemann-Stieltjes Integral //385
APPENDIX C Notation and Abbreviations //389
Selected Referel1ces //391
Index //393