Probability for Statisticians

Preface......Page 8
Contents......Page 10
Use of this Text......Page 14
Definition of Symbols......Page 19
1 Basic Properties of Measures......Page 20
2 Construction and Extension of Measures......Page 31
3 Lebesgue-Stieltjes Measures......Page 37
1 Mappings and  igma-Fields......Page 40
2 Measurable Functions......Page 43
3 Convergence......Page 48
4 Probability, RVs, and Convergence in Law......Page 52
5 Discussion of Sub  igma-Fields......Page 54
1 The Lebesgue Integral......Page 56
2 Fundamental Properties of Integrals......Page 59
3 Evaluating and Differentiating Integrals......Page 63
4 Inequalities......Page 65
5 Modes of Convergence......Page 70
1 Decomposition of Signed Measures......Page 80
2 The Radon-Nikodym Theorem......Page 85
3 Lebesgue\'s Theorem......Page 89
4 The Fundamental Theorem of Calculus......Page 93
1 Finite-Dimensional Product Measures......Page 98
2 Random Vectors on (\\Omega, \\cal{A}, P)......Page 103
3 Countably Infinite Product Probability Spaces......Page 105
4 Random Elements and Processes on (\\Omega, \\cal{A}, P)......Page 109
1 General Topology......Page 114
2 Metric Spaces......Page 120
3 Hilbert Space......Page 123
1 Character of Distribution Functions......Page 126
2 Properties of Distribution Functions......Page 129
3 The Quantile Transformation......Page 130
4 Integration by Parts Applied to Moments......Page 134
5 Important Statistical Quantities......Page 138
6 Infinite Variance......Page 142
7 Slowly Varying Partial Variance......Page 146
8 Specific Tail Relationships......Page 153
9 Regularly Varying Functions......Page 156
10 Some Winsorized Variance Comparisons......Page 159
11 Inequalities for Winsorized Quantile Functions......Page 166
1 Independence......Page 170
2 The Tail  igma-Field......Page 174
3 Uncorrelated Random Variables......Page 176
4 Basic Properties of Conditional Expectation......Page 177
5 Regular Conditional Probability......Page 187
6 Conditional Expectations as Projections......Page 193
1 Elementary Probability......Page 198
2 Distribution Theory for Statistics......Page 206
3 Linear Algebra Applications......Page 210
4 The Multivariate Normal Distribution......Page 218
0 Introduction......Page 222
1 Borel-Cantelli and Kronecker lemmas......Page 223
2 Truncation, WLLN, and Review of Inequalities......Page 225
3 Maximal Inequalities and Symmetrization......Page 229
4 The Classical Laws of Large Numbers, LLNs......Page 234
5 Applications of the Laws of Large Numbers......Page 242
6 General Moment Estimation......Page 245
7 Law of the Iterated Logarithm......Page 254
8 Strong Markov Property for Sums of IID RVs......Page 258
9 Convergence of Series of Independent RVs......Page 260
10 Martingales......Page 265
11 Maximal Inequalities, Some with Boundaries......Page 266
12 A Uniform SLLN......Page 271
1 Steins Method for CLTs......Page 274
2 Winsorization and Truncation......Page 283
3 Identically Distributed RVs......Page 288
4 Bootstrapping......Page 293
5 Bootstrapping with Slowly Increasing Trimming......Page 295
6 Examples of Limiting Distributions......Page 298
7 Classical Convergence in Distribution......Page 307
8 Limit Determining Classes of Functions......Page 311
1 Special Spaces......Page 314
2 Existence of Processes on (C, \\cal{C}) and (D, \\cal{D})......Page 317
3 Brownian Motion and Brownian Bridge......Page 321
4 Stopping Times......Page 324
5 Strong Markov Property......Page 327
6 Embedding a RV in Brownian Motion......Page 330
7 Barrier Crossing Probabilities......Page 333
8 Embedding the Partial Sum Process......Page 337
9 Other Properties of Brownian Motion......Page 342
10 Various Empirical Process......Page 344
11 Inequalities for Various Empirical Processes......Page 352
12 Applications......Page 357
1 Basic Results, with Derivation of Common Chfs......Page 360
2 Uniqueness and Inversion......Page 365
3 The Continuity Theorem......Page 369
4 Elementary Complex and Fourier Analysis......Page 371
5 Esseen s Lemma......Page 377
6 Distributions on Grids......Page 380
7 Conditions for f to Be a Characteristic Function......Page 382
0 Introduction......Page 384
1 Basic Limit Theorems......Page 385
2 Variations on the Classical CLT......Page 390
3 Local Limit Theorems......Page 399
4 Gamma Approximation......Page 402
5 Edgeworth Expansions......Page 409
6 Approximating the Distribution of h(\\bar{X}_n)......Page 415
1 Infinitely Divisible Distributions......Page 418
2 Stable Distributions......Page 426
3 Characterizing Stable Laws......Page 429
4 The Domain of Attraction of a Stable Law......Page 431
0 Introduction......Page 434
1 Trimmed and Winsorized Means......Page 435
2 Linear Rank Statistics and Finite Sampling......Page 445
3 The Bootstrap......Page 451
4 L-Statistics......Page 456
1 U-Statistics......Page 468
2 Hoeffding\'s Combinatorial CLT......Page 477
1 Basic Technicalities for Martingales......Page 486
2 Simple Optional Sampling Theorem......Page 491
3 The Submartingale Convergence Theorem......Page 492
4 Applications of the S-mg Convergence Theorem......Page 500
5 Decomposition of a Submartingale Sequence......Page 506
6 Optional Sampling......Page 511
7 Applications of Optional Sampling......Page 518
8 Introduction to Counting Process Martingales......Page 520
9 Doob-Meyer Submartingale Decomposition......Page 530
10 Predictable Processes and \\int H dM Martingales......Page 535
11 The Basic Censored Data Martingale......Page 541
12 CLTs for Dependent RVs......Page 548
1 Convergence in Distribution on Metric Spaces......Page 550
2 Metrics for Convergence in Distribution......Page 559
A Distribution Summaries......Page 564
1 The Gamma and Digamma Functions......Page 565
2 Maximum Likelihood Estimators and Moments......Page 570
3 Examples of Statistical Models......Page 574
4 Asymptotics of Maximum Likelihood Estimation......Page 582
References......Page 587
Index......Page 594