Uniform central limit theorems:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Edition: | Digitally printed version |
| Series: | Cambridge studies in advanced mathematics
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 436 S. graph. Darst. |
| ISBN: | 9780521052214 |
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MARC
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| 100 | 1 | |a Dudley, Richard M. |d 1938- |e Verfasser |0 (DE-588)121010996 |4 aut | |
| 245 | 1 | 0 | |a Uniform central limit theorems |c R. M. Dudley |
| 250 | |a Digitally printed version | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2008 | |
| 300 | |a XIV, 436 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 63 | |
| 650 | 4 | |a Central limit theorem | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-020525563 | ||
Record in the Search Index
| _version_ | 1804143208518647808 |
|---|---|
| adam_text | Contents
Preface poge
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
χ
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-Červonenkis
Combinatorics
134
4.1
Vapnik-Červonenkis
classes
134
4.2
Generating
Vapnik-Červonenkis
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-Červonenkis
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-Červonenkis
and Related Classes
1%
6.1
Koltchinskii-Pollard entropy and Glivenko-Cantelli theorems
196
6.2
Vapnik-Červonenkis-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 Hubert 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
Giné-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
В
Multinomial Distributions
399
Appendix
С
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;
Borei
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
|
| any_adam_object | 1 |
| author | Dudley, Richard M. 1938- |
| author_GND | (DE-588)121010996 |
| author_facet | Dudley, Richard M. 1938- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 21 |
| dewey-search | 519.2 21 |
| dewey-sort | 3519.2 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Digitally printed version |
| format | Book |
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| id | DE-604.BV036605158 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:44:01Z |
| institution | BVB |
| isbn | 9780521052214 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020525563 |
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| physical | XIV, 436 S. graph. Darst. |
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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 Digitally printed version Cambridge [u.a.] Cambridge Univ. Press 2008 XIV, 436 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 63 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 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020525563&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 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 | Central limit theorem Zentraler Grenzwertsatz (DE-588)4067618-3 gnd |
| topic_facet | Central limit theorem Zentraler Grenzwertsatz |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020525563&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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