Normal Approximations with Malliavin Calculus

From Stein's Method to Universality

AvIvan Nourdin,Giovanni Peccati

Inbunden, Engelska, 2012

Del 192 i serien Cambridge Tracts in Mathematics

995 kr

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Beskrivning

Stein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, Breuer-Major theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wish to understand the connections between Stein's method and Malliavin calculus.

Produktinformation

Utgivningsdatum:2012-05-10
Mått:152 x 229 x 18 mm
Vikt:490 g
Format:Inbunden
Språk:Engelska
Serie:Cambridge Tracts in Mathematics
Antal sidor:254
Förlag:Cambridge University Press
ISBN:9781107017771
Utmärkelser:Outstanding Scientific Publication Prize, National Foundation for Science of Luxembourg 2015

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Mer om författaren

Ivan Nourdin is Full Professor at Nancy University 1, France. Giovanni Peccati is Full Professor in Stochastic Analysis and Finance at the University of Luxembourg.

Recensioner i media

'This monograph is a nice and excellent introduction to Malliavin calculus and its application to deducing quantitative central limit theorems in combination with Stein's method for normal approximation. It provides a self-contained and appealing presentation of the recent work developed by the authors, and it is well tailored for graduate students and researchers.' David Nualart, Mathematical Reviews

Innehållsförteckning

Preface; Introduction; 1. Malliavin operators in the one-dimensional case; 2. Malliavin operators and isonormal Gaussian processes; 3. Stein's method for one-dimensional normal approximations; 4. Multidimensional Stein's method; 5. Stein meets Malliavin: univariate normal approximations; 6. Multivariate normal approximations; 7. Exploring the Breuer–Major Theorem; 8. Computation of cumulants; 9. Exact asymptotics and optimal rates; 10. Density estimates; 11. Homogeneous sums and universality; Appendix 1. Gaussian elements, cumulants and Edgeworth expansions; Appendix 2. Hilbert space notation; Appendix 3. Distances between probability measures; Appendix 4. Fractional Brownian motion; Appendix 5. Some results from functional analysis; References; Index.