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