Preface ........................................................ xi
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: Hermite polynomials ................. 13
1.5 Second application: variance expansions ................ 15
1.6 Third application: second-order Poincare 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 Hilbert spaces ............ 32
2.5 The divergence operator ................................ 33
2.6 Some Hilbert 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 Stein'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
4 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
5 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
6 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
7 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
8 Computation of cumulants ................................... 148
8.1 Decomposing multi-indices ............................. 148
8.2 General formulae ...................................... 149
8.3 Application to multiple integrals ..................... 154
8.4 Formulae in dimension one ............................. 151
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.l 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.l General notation ...................................... 205
B.2 L2 spaces ............................................. 205
B.3 More on symmetrization ................................ 205
B.4 Contractions .......................................... 206
B.5 Random elements ....................................... 208
B.6 Bibliographic comments ................................ 208
Appendix С Distances between probability measures ............ 209
C.l General definitions ................................... 209
C.2 Some special distances ................................ 210
C.3 Some further results .................................. 211
C.4 Bibliographic comments ................................ 214
Appendix D Fractional Brownian motion ........................ 215
D.l 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.l 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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