包邮测度理论概率导论

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—Jordan Decomposition 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 10 Independence ／／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 Lemma 1 in Chapter 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