Uniform central limit theorems

Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Dedication......Page 8
Contents......Page 10
Preface......Page 14
1 Introduction: Donsker's Theorem, Metric Entropy,  and Inequalities......Page 16
1.1 Empirical processes: the classical case......Page 17
1.2 Metric entropy and capacity......Page 25
1.3 Inequalities......Page 27
Problems......Page 33
Notes......Page 34
References......Page 36
2.1 Some definitions......Page 38
2.2 Gaussian vectors are probably not very large......Page 39
2.3 Inequalities and comparisons for Gaussian distributions......Page 46
2.4 Gaussian measures and convexity......Page 55
2.5 The isonormal process: sample boundedness and continuity......Page 58
2.6 A metric entropy sufficient condition for sample continuity......Page 67
2.7 Majorizing measures......Page 74
2.8 Sample continuity and compactness......Page 89
**2.9 Volumes, mixed volumes, and ellipsoids......Page 93
**2.10 Convex hulls of sequences......Page 97
Problems......Page 98
Notes......Page 101
References......Page 103
3.1 Definitions: convergence in law......Page 106
3.2 Measurable cover functions......Page 110
3.3 Almost uniform convergence amd convergence in  outer probability......Page 115
3.4 Perfect functions......Page 118
3.5 Almost surely convergent realizations......Page 121
3.6 Conditions equivalent to convergence in law......Page 126
3.7 Asymptotic equicontinuity and Donsker classes......Page 132
3.8 Unions of Donsker classes......Page 136
3.9 Sequences of sets and functions......Page 137
Problems......Page 142
Notes......Page 145
References......Page 147
4.1 Vapnik-Cervonenkis classes......Page 149
4.2 Generating Vapnik-Cervonenkis classes......Page 153
*4.3 Maximal classes......Page 157
*4.4 Classes of index 1......Page 160
*4.5 Combining VC classes......Page 167
4.6 Probability laws and independence......Page 171
4.7 Vapnik-Cervonenkis properties of classes of functions......Page 174
4.8 Classes of functions and dual density......Page 176
**4.9 Further facts about VC classes......Page 180
Problems......Page 181
Notes......Page 182
References......Page 183
5 Measurability......Page 185
*5.1 Sufficiency......Page 186
5.2 Admissibility......Page 194
5.3 Suslin properties, selection, and a counterexample......Page 200
Problems......Page 206
Notes......Page 208
References......Page 209
6.1 Koltchinskii-Pollard entropy and Glivenko-Cantelli theorems......Page 211
6.2 Vapnik-Cervonenkis-Steele laws of large numbers......Page 218
6.3 Pollard's central limit theorem......Page 223
6.4 Necessary conditions for limit theorems......Page 230
**6.5 Inequalities for empirical processes......Page 235
**6.6 Glivenko-Cantelli properties and random entropy......Page 238
**6.7 Classification problems and learning theory......Page 241
Problems......Page 242
Notes......Page 243
References......Page 245
7.1 Definitions and the Blum-DeHardt law of large numbers......Page 249
7.2 Central limit theorems with bracketing......Page 253
7.3 The power set of a countable set: the Borisov-Durst theorem......Page 259
**7.4 Bracketing and majorizing measures......Page 261
Problems......Page 262
References......Page 263
8.1 Introduction: the Hausdorff metric......Page 265
8.2 Spaces of differentiable functions and sets with differentiable boundaries......Page 267
8.3 Lower layers......Page 279
8.4 Metric entropy of classes of convex sets......Page 284
Problems......Page 296
Notes......Page 297
References......Page 298
9 Sums in General Banach Spaces and Invariance Principles......Page 300
9.1 Independent random elements and partial sums......Page 301
9.2 A CLT implies measurability in separable normed spaces......Page 306
9.3 A finite-dimensional invariance principle......Page 308
9.4 Invariance principles for empirical processes......Page 316
**9.5 Log log laws and speeds of convergence......Page 321
Problems......Page 324
Notes......Page 325
References......Page 326
10.1 Universal Donsker classes......Page 329
10.2 Metric entropy of convex hulls in Hilbert space......Page 337
**10.3 Uniform Donsker classes......Page 343
References......Page 345
11.1 The two-sample case......Page 347
11.2 A bootstrap central limit theorem in probability......Page 350
11.3 Other aspects of the bootstrap......Page 372
** 11.4 Further Gine-Zinn bootstrap central limit theorems......Page 373
Problems......Page 374
Notes......Page 375
References......Page 376
12.1 Universal lower bounds......Page 378
12.2 An upper bound......Page 380
12.3 Poissonization and random sets......Page 382
12.4 Lower bounds in borderline cases......Page 388
12.5 Proof of Theorem 12.4.1......Page 399
Notes......Page 403
References......Page 404
Appendix A Differentiating under an Integral Sign......Page 406
Appendix B Multinomial Distributions......Page 414
Appendix C Measures on Nonseparable Metric Spaces......Page 417
Appendix D An Extension of Lusin's Theorem......Page 420
Appendix E Bochner and Pettis Integrals......Page 422
Appendix F Nonexistence of Types of Linear Forms on Some Spaces......Page 428
Appendix G Separation of Analytic Sets; Borel Injections......Page 432
Appendix H Young-Orlicz Spaces......Page 436
Appendix I Modifications and Versions of Isonormal Processes......Page 440
Subject Index......Page 442
Author Index......Page 447
Index of Notation......Page 450