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Normal Approximations with Malliavin Calculus : From Stein's Method to Universality.
Shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2012.
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| Colección: | Cambridge Tracts in Mathematics, 192.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; CAMBRIDGE TRACTS IN MATHEMATICS: GENERAL EDITORS; Title; Copyright; Dedication; Contents; Preface; Introduction; 1 Malliavin operators in the one-dimensional case; 1.1 Derivative operators; 1.2 Divergences; 1.3 Ornstein-Uhlenbeck operators; 1.4 First application: Hermite polynomials; 1.5 Second application: variance expansions; 1.6 Third application: second-order Poincaré inequalities; 1.7 Exercises; 1.8 Bibliographic comments; 2 Malliavin operators and isonormal Gaussian processes; 2.1 Isonormal Gaussian processes; 2.2 Wiener chaos; 2.3 The derivative operator.
- 2.4 The Malliavin derivatives in Hilbert spaces2.5 The divergence operator; 2.6 Some Hilbert space valued divergences; 2.7 Multiple integrals; 2.8 The Ornstein-Uhlenbeck semigroup; 2.9 An integration by parts formula; 2.10 Absolute continuity of the laws of multiple integrals; 2.11 Exercises; 2.12 Bibliographic comments; 3 Stein's method for one-dimensional normal approximations; 3.1 Gaussian moments and Stein's lemma; 3.2 Stein's equations; 3.3 Stein's bounds for the total variation distance; 3.4 Stein's bounds for the Kolmogorov distance; 3.5 Stein's bounds for the Wasserstein distance.
- 3.6 A simple example3.7 The Berry-Esseen theorem; 3.8 Exercises; 3.9 Bibliographic comments; 4 Multidimensional Stein's method; 4.1 Multidimensional Stein's lemmas; 4.2 Stein's equations for identity matrices; 4.3 Stein's equations for general positive definite matrices; 4.4 Bounds on the Wasserstein distance; 4.5 Exercises; 4.6 Bibliographic comments; 5 Stein meets Malliavin: univariate normal approximations; 5.1 Bounds for general functionals; 5.2 Normal approximations on Wiener chaos; 5.2.1 Some preliminary considerations; 5.3 Normal approximations in the general case; 5.3.1 Main results.
- 5.4 Exercises5.5 Bibliographic comments; 6 Multivariate normal approximations; 6.1 Bounds for general vectors; 6.2 The case of Wiener chaos; 6.3 CLTs via chaos decompositions; 6.4 Exercises; 6.5 Bibliographic comments; 7 Exploring the Breuer-Major theorem; 7.1 Motivation; 7.2 A general statement; 7.3 Quadratic case; 7.4 The increments of a fractional Brownian motion; 7.5 Exercises; 7.6 Bibliographic comments; 8 Computation of cumulants; 8.1 Decomposing multi-indices; 8.2 General formulae; 8.3 Application to multiple integrals; 8.4 Formulae in dimension one; 8.5 Exercises.
- 8.6 Bibliographic comments9 Exact asymptotics and optimal rates; 9.1 Some technical computations; 9.2 A general result; 9.3 Connections with Edgeworth expansions; 9.4 Double integrals; 9.5 Further examples; 9.6 Exercises; 9.7 Bibliographic comments; 10 Density estimates; 10.1 General results; 10.2 Explicit computations; 10.3 An example; 10.4 Exercises; 10.5 Bibliographic comments; 11 Homogeneous sums and universality; 11.1 The Lindeberg method; 11.2 Homogeneous sums and influence functions; 11.3 The universality result; 11.4 Some technical estimates; 11.5 Proof of Theorem 11.3.1.