Normal approximations with Malliavin calculus

CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Introduction......Page 17
 1.1 Derivative operators......Page 20
 1.2 Divergences......Page 24
 1.3 Ornstein–Uhlenbeck operators......Page 25
 1.4 First application: Hermite polynomials......Page 29
 1.5 Second application: variance expansions......Page 31
 1.6 Third application: second-order Poincaré inequalities......Page 32
 1.7 Exercises......Page 35
 1.8 Bibliographic comments......Page 36
 2.1 Isonormal Gaussian processes......Page 38
 2.2 Wiener chaos......Page 42
 2.3 The derivative operator......Page 44
 2.4 The Malliavin derivatives in Hilbert spaces......Page 48
 2.5 The divergence operator......Page 49
 2.6 Some Hilbert space valued divergences......Page 51
 2.7 Multiple integrals......Page 52
 2.8 The Ornstein–Uhlenbeck semigroup......Page 61
 2.9 An integration by parts formula......Page 69
 2.10 Absolute continuity of the laws of multiple integrals......Page 70
 2.11 Exercises......Page 71
 2.12 Bibliographic comments......Page 73
 3.1 Gaussian moments and Stein's lemma......Page 75
 3.2 Stein's equations......Page 78
 3.3 Stein's bounds for the total variation distance......Page 79
 3.4 Stein's bounds for the Kolmogorov distance......Page 81
 3.5 Stein's bounds for the Wasserstein distance......Page 83
 3.6 A simple example......Page 85
 3.7 The Berry–Esseen theorem......Page 86
 3.8 Exercises......Page 91
 3.9 Bibliographic comments......Page 94
 4.1 Multidimensional Stein's lemmas......Page 95
 4.2 Stein's equations for identity matrices......Page 97
 4.3 Stein's equations for general positive definite matrices......Page 100
 4.4 Bounds on the Wasserstein distance......Page 101
 4.5 Exercises......Page 102
 4.6 Bibliographic comments......Page 104
 5.1 Bounds for general functionals......Page 105
  5.2.1 Some preliminary considerations......Page 109
  5.3.1 Main results......Page 118
 5.4 Exercises......Page 124
 5.5 Bibliographic comments......Page 131
 6.1 Bounds for general vectors......Page 132
 6.2 The case of Wiener chaos......Page 136
 6.3 CLTs via chaos decompositions......Page 140
 6.4 Exercises......Page 142
 6.5 Bibliographic comments......Page 143
 7.1 Motivation......Page 144
 7.2 A general statement......Page 145
 7.3 Quadratic case......Page 149
 7.4 The increments of a fractional Brownian motion......Page 154
 7.5 Exercises......Page 161
 7.6 Bibliographic comments......Page 162
 8.1 Decomposing multi-indices......Page 164
 8.2 General formulae......Page 165
 8.3 Application to multiple integrals......Page 170
 8.4 Formulae in dimension one......Page 173
 8.6 Bibliographic comments......Page 175
 9.1 Some technical computations......Page 176
 9.2 A general result......Page 177
 9.3 Connections with Edgeworth expansions......Page 179
 9.4 Double integrals......Page 181
 9.5 Further examples......Page 182
 9.6 Exercises......Page 184
 9.7 Bibliographic comments......Page 185
 10.1 General results......Page 186
 10.2 Explicit computations......Page 190
 10.3 An example......Page 191
 10.4 Exercises......Page 192
 10.5 Bibliographic comments......Page 194
 11.1 The Lindeberg method......Page 195
 11.2 Homogeneous sums and influence functions......Page 198
 11.3 The universality result......Page 201
 11.4 Some technical estimates......Page 204
 11.5 Proof of Theorem 11.3.1......Page 210
 11.6 Exercises......Page 211
 11.7 Bibliographic comments......Page 212
 A.1 Gaussian random variables......Page 213
 A.2 Cumulants......Page 214
 A.3 The method of moments and cumulants......Page 218
 A.4 Edgeworth expansions in dimension one......Page 219
 A.5 Bibliographic comments......Page 220
 B.3 More on symmetrization......Page 221
 B.4 Contractions......Page 222
 B.6 Bibliographic comments......Page 224
 C.1 General definitions......Page 225
 C.2 Some special distances......Page 226
 C.3 Some further results......Page 227
 C.4 Bibliographic comments......Page 230
 D.1 Definition and immediate properties......Page 231
 D.2 Hurst phenomenon and invariance principle......Page 234
 D.3 Fractional Brownian motion is not a semimartingale......Page 237
 D.4 Bibliographic comments......Page 240
 E.1 Dense subsets of an Lq space......Page 241
 E.3 Bibliographic comments......Page 242
References......Page 243
Author index......Page 251
Notation index......Page 253
Subject index......Page 254