Normal approximations with Malliavin calculus: from Stein's method to universality
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2012
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
192 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIV, 239 S. |
| ISBN: | 9781107017771 hbk 1107017777 |
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MARC
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| 100 | 1 | |a Nourdin, Ivan |d 1978- |e Verfasser |0 (DE-588)1023866684 |4 aut | |
| 245 | 1 | 0 | |a Normal approximations with Malliavin calculus |b from Stein's method to universality |c Ivan Nourdin ; Giovanni Peccati |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2012 | |
| 300 | |a XIV, 239 S. | ||
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| 490 | 1 | |a Cambridge tracts in mathematics |v 192 | |
| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Malliavin calculus | |
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Record in the Search Index
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| adam_text | Contents
Preface VaEe
χί
Introduction
1
1
Malliavin
operators
in the one-dimensional case
4
1.1
Derivative operators
4
1.2
Divergences
8
1.3
Ornstein-Uhlenbeck operators
9
1.4
First application: Hermi
te
polynomials
13
1.5
Second application: variance expansions
15
1.6
Third application: second-order
Poincaré
inequalities
16
1.7
Exercises
19
1.8
Bibliographic comments
20
2
Malliavin
operators and
isonormal
Gaussian processes
22
2.1
Isonormal
Gaussian processes
22
2.2
Wiener chaos
26
2.3
The derivative operator
28
2.4
The Malliavin derivatives in Hubert spaces
32
2.5
The divergence operator
33
2.6
Some Hubert space valued divergences
35
2.7
Multiple integrals
36
2.8
The Ornstein-Uhlenbeck semigroup
45
2.9
An integration by parts formula
53
2.10
Absolute continuity of the laws of multiple integrals
54
2.11
Exercises
55
2.12
Bibliographic comments
57
3
Stem s method for one-dimensional normal approximations
59
3.1
Gaussian moments and Stein s lemma
59
3.2
Stein s equations
62
3.3
Stein s bounds for the total variation distance
63
3.4
Stein s bounds for the Kolmogorov distance
65
3.5
Stein s bounds for the
Wasserstein
distance
67
3.6
A simple example
69
3.7
The Berry-Esseen theorem
70
3.8
Exercises
75
3.9
Bibliographic comments
78
Multidimensional
Stein s method
79
4.1
Multidimensional Stein s lemmas
79
4.2
Stein s equations for identity matrices
81
4.3
Stein s equations for general positive definite matrices
84
4.4
Bounds on the
Wasserstein
distance
85
4.5
Exercises
86
4.6
Bibliographic comments
88
Stein meets Malliavin: univariate normal approximations
89
5.1
Bounds for general functionals
89
5.2
Normal approximations on Wiener chaos
93
5.3
Normal approximations in the general case
102
5.4
Exercises
108
5.5
Bibliographic comments
115
Multivariate normal approximations
116
6.1
Bounds for general vectors
116
6.2
The case of Wiener chaos
120
6.3
CLTs via chaos decompositions
124
6.4
Exercises
126
6.5
Bibliographic comments
127
Exploring the
Breuer-Major
theorem
128
7.1
Motivation
128
7.2
A general statement
129
7.3
Quadratic case
133
7.4
The increments of a fractional Brownian motion
138
7.5
Exercises
145
7.6
Bibliographic comments
146
Computation of
cumulants
148
8.1
Decomposing multi-indices
148
8.2
General formulae
149
8.3
AüDlication
to multiple integrals
154
8.4
Formulae in dimension one
157
8.5
Exercises
159
8.6
Bibliographic comments
159
9
Exact asymptotics and optimal rates
160
9.1
Some technical computations
160
9.2
A general result
161
9.3
Connections with Edgeworth expansions
163
9.4
Double integrals
165
9.5
Further examples
166
9.6
Exercises
168
9.7
Bibliographic comments
169
10
Density estimates
170
10.1
General results
170
10.2
Explicit computations
174
10.3
An example
175
10.4
Exercises
176
10.5
Bibliographic comments
178
11
Homogeneous sums and universality
179
11.1
The
Lindeberg
method
179
11.2
Homogeneous sums and influence functions
182
11.3
The universality result
185
11.4
Some technical estimates
188
11.5
Proof of Theorem
11.3.1 194
11.6
Exercises
195
11.7
Bibliographic comments
196
Appendix A Gaussian elements,
cumulants
and Edgeworth
expansions
197
A.I Gaussian random variables
197
A.2
Cumulants
198
A.3 The method of moments and
cumulants
202
A.4 Edgeworth expansions in dimension one
203
A.5 Bibliographic comments
204
Appendix
В
Hilbert space notation
205
B.I General notation
205
B.2 L2 spaces
205
B.3 More on symmetrization
205
B.4 Contractions
206
χ
Contents
8.
5
Random elements
208
8.
6
Bibliographic comments
208
Appendix
С
Distances between probability measures
209
C.I General definitions
209
C.2 Some special distances
210
C.3 Some further results
211
C.4 Bibliographic comments
214
Appendix
D
Fractional Browman motion
215
D.I Definition and immediate properties
215
D.2 Hurst phenomenon and
invariance
principle
218
D.3 Fractional Brownian motion is not a semimartingale
221
D.4 Bibliographic comments
224
Appendix
E Some
results from functional analysis
225
E.I Dense subsets of an Lq space
225
E.2 Rademacher s theorem
226
E.3 Bibliographic comments
226
References
227
Author index
235
Notation index
237
Subject index
238
|
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| spelling | Nourdin, Ivan 1978- Verfasser (DE-588)1023866684 aut Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin ; Giovanni Peccati 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2012 XIV, 239 S. txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 192 Includes bibliographical references and index Approximation theory Malliavin calculus MATHEMATICS / Probability & Statistics / General bisacsh Malliavin-Kalkül (DE-588)4242584-0 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Stein-Schätzung (DE-588)7570767-6 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd rswk-swf Wahrscheinlichkeitsverteilung (DE-588)4121894-2 s Approximation (DE-588)4002498-2 s Malliavin-Kalkül (DE-588)4242584-0 s Stein-Schätzung (DE-588)7570767-6 s DE-604 Peccati, Giovanni 1975- Verfasser (DE-588)1018525807 aut Cambridge tracts in mathematics 192 (DE-604)BV000000001 192 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024842729&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nourdin, Ivan 1978- Peccati, Giovanni 1975- Normal approximations with Malliavin calculus from Stein's method to universality Cambridge tracts in mathematics Approximation theory Malliavin calculus MATHEMATICS / Probability & Statistics / General bisacsh Malliavin-Kalkül (DE-588)4242584-0 gnd Approximation (DE-588)4002498-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| subject_GND | (DE-588)4242584-0 (DE-588)4002498-2 (DE-588)7570767-6 (DE-588)4121894-2 |
| title | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_auth | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_exact_search | Normal approximations with Malliavin calculus from Stein's method to universality |
| title_full | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin ; Giovanni Peccati |
| title_fullStr | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin ; Giovanni Peccati |
| title_full_unstemmed | Normal approximations with Malliavin calculus from Stein's method to universality Ivan Nourdin ; Giovanni Peccati |
| title_short | Normal approximations with Malliavin calculus |
| title_sort | normal approximations with malliavin calculus from stein s method to universality |
| title_sub | from Stein's method to universality |
| topic | Approximation theory Malliavin calculus MATHEMATICS / Probability & Statistics / General bisacsh Malliavin-Kalkül (DE-588)4242584-0 gnd Approximation (DE-588)4002498-2 gnd Stein-Schätzung (DE-588)7570767-6 gnd Wahrscheinlichkeitsverteilung (DE-588)4121894-2 gnd |
| topic_facet | Approximation theory Malliavin calculus MATHEMATICS / Probability & Statistics / General Malliavin-Kalkül Approximation Stein-Schätzung Wahrscheinlichkeitsverteilung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024842729&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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