Uniform central limit theorems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1999
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
63 |
| Subjects: | |
| Online Access: | Sample text Table of contents Publisher description Inhaltsverzeichnis |
| Physical Description: | XIV, 436 S. graph. Darst. |
| ISBN: | 0521461022 |
Staff View
MARC
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| 245 | 1 | 0 | |a Uniform central limit theorems |c R. M. Dudley |
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Record in the Search Index
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| adam_text | Contents
Preface page xiii
1 Introduction: Donsker s Theorem, Metric Entropy,
and Inequalities 1
1.1 Empirical processes: the classical case 2
1.2 Metric entropy and capacity 10
1.3 Inequalities 12
Problems 18
Notes 19
References 21
2 Gaussian Measures and Processes; Sample Continuity 23
2.1 Some definitions 23
2.2 Gaussian vectors are probably not very large 24
2.3 Inequalities and comparisons for Gaussian distributions 31
2.4 Gaussian measures and convexity 40
2.5 The isonormal process: sample boundedness and continuity 43
2.6 A metric entropy sufficient condition for sample continuity 52
2.7 Majorizing measures 59
2.8 Sample continuity and compactness 74
**2.9 Volumes, mixed volumes, and ellipsoids 78
**2.10 Convex hulls of sequences 82
Problems 83
Notes 86
References 88
3 Foundations of Uniform Central Limit Theorems:
Donsker Classes 91
3.1 Definitions: convergence in law 91
3.2 Measurable cover functions 95
ix
x Contents
3.3 Almost uniform convergence amd convergence in
outer probability 100
3.4 Perfect functions 103
3.5 Almost surely convergent realizations 106
3.6 Conditions equivalent to convergence in law 111
3.7 Asymptotic equicontinuity and Donsker classes 117
3.8 Unions of Donsker classes 121
3.9 Sequences of sets and functions 122
Problems 127
Notes 130
References 132
4 Vapnik Cervonenkis Combinatorics 134
4.1 Vapnik Cervonenkis classes 134
4.2 Generating Vapnik Cervonenkis classes 138
*4.3 Maximal classes 142
*4.4 Classes of index 1 145
*4.5 Combining VC classes 152
4.6 Probability laws and independence 156
4.7 Vapnik Cervonenkis properties of classes of functions 159
4.8 Classes of functions and dual density 161
**4.9 Further facts about VC classes 165
Problems 166
Notes 167
References 168
5 Measurability 170
*5.1 Sufficiency 171
5.2 Admissibility 179
5.3 Suslin properties, selection, and a counterexample 185
Problems 191
Notes 193
References 194
6 Limit Theorems for Vapnik Cervonenkis and Related Classes 196
6.1 Koltchinskii Pollard entropy and Glivenko Cantelli theorems 196
6.2 Vapnik Cervonenkis Steele laws of large numbers 203
6.3 Pollard s central limit theorem 208
6.4 Necessary conditions for limit theorems 215
**6.5 Inequalities for empirical processes 220
**6.6 Glivenko Cantelli properties and random entropy 223
**6.7 Classification problems and learning theory 226
Problems 227
Contents xi
Notes 228
References 230
7 Metric Entropy, with Inclusion and Bracketing 234
7.1 Definitions and the Blum DeHardt law of large numbers 234
7.2 Central limit theorems with bracketing 238
7.3 The power set of a countable set: the Borisov Durst theorem 244
**7.4 Bracketing and majorizing measures 246
Problems 247
Notes 248
References 248
8 Approximation of Functions and Sets 250
8.1 Introduction: the Hausdorff metric 250
8.2 Spaces of differentiable functions and sets with differentiable
boundaries 252
8.3 Lower layers 264
8.4 Metric entropy of classes of convex sets 269
Problems 281
Notes 282
References 283
9 Sums in General Banach Spaces and Invariance Principles 285
9.1 Independent random elements and partial sums 286
9.2 A CLT implies measurability in separable normed spaces 291
9.3 A finite dimensional invariance principle 293
9.4 Invariance principles for empirical processes 301
**9.5 Log log laws and speeds of convergence 306
Problems 309
Notes 310
References 311
10 Universal and Uniform Central Limit Theorems 314
10.1 Universal Donsker classes 314
10.2 Metric entropy of convex hulls in Hilbert space 322
**10.3 Uniform Donsker classes 328
Problems 330
Notes 330
References 330
11 The Two Sample Case, the Bootstrap, and Confidence Sets 332
11.1 The two sample case 332
11.2 A bootstrap central limit theorem in probability 335
11.3 Other aspects of the bootstrap 357
xii Contents
**11.4 Further Gine Zinn bootstrap central limit theorems 358
Problems 359
Notes 360
References 361
12 Classes of Sets or Functions Too Large for Central
Limit Theorems 363
12.1 Universal lower bounds 363
12.2 An upper bound 365
12.3 Poissonization and random sets 367
12.4 Lower bounds in borderline cases 373
12.5 Proof of Theorem 12.4.1 384
Problems 388
Notes 388
References 389
Appendix A Differentiating under an Integral Sign 391
Appendix B Multinomial Distributions 399
Appendix C Measures on Nonseparable Metric Spaces 402
Appendix D An Extension of Lusin s Theorem 405
Appendix E Bochner and Pettis Integrals 407
Appendix F Nonexistence of Types of Linear Forms on Some Spaces 413
Appendix G Separation of Analytic Sets; Borel Injections 417
Appendix H Young Orlicz Spaces 421
Appendix I Modifications and Versions of Isonormal Processes 425
Subject Index 427
Author Index 432
Index of Notation 435
|
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| isbn | 0521461022 |
| language | English |
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| series2 | Cambridge studies in advanced mathematics |
| spelling | Dudley, Richard M. 1938- Verfasser (DE-588)121010996 aut Uniform central limit theorems R. M. Dudley 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1999 XIV, 436 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 63 Zentraler Grenzwertsatz Central limit theorem Zentraler Grenzwertsatz (DE-588)4067618-3 gnd rswk-swf Zentraler Grenzwertsatz (DE-588)4067618-3 s DE-604 Cambridge studies in advanced mathematics 63 (DE-604)BV000003678 63 http://www.loc.gov/catdir/samples/cam032/98035562.html Sample text http://www.loc.gov/catdir/toc/cam024/98035562.html Table of contents http://www.loc.gov/catdir/description/cam029/98035562.html Publisher description HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008599603&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dudley, Richard M. 1938- Uniform central limit theorems Cambridge studies in advanced mathematics Zentraler Grenzwertsatz Central limit theorem Zentraler Grenzwertsatz (DE-588)4067618-3 gnd |
| subject_GND | (DE-588)4067618-3 |
| title | Uniform central limit theorems |
| title_auth | Uniform central limit theorems |
| title_exact_search | Uniform central limit theorems |
| title_full | Uniform central limit theorems R. M. Dudley |
| title_fullStr | Uniform central limit theorems R. M. Dudley |
| title_full_unstemmed | Uniform central limit theorems R. M. Dudley |
| title_short | Uniform central limit theorems |
| title_sort | uniform central limit theorems |
| topic | Zentraler Grenzwertsatz Central limit theorem Zentraler Grenzwertsatz (DE-588)4067618-3 gnd |
| topic_facet | Zentraler Grenzwertsatz Central limit theorem |
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