Normal Approximations with Malliavin Calculus

From Stein's Method to Universality

By: Ivan Nourdin, Giovanni Peccati

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Hardcover | 10 May 2012 | Edition Number 1

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254 Pages

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22.86 x 15.24 x 1.75

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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, Breuerâ"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.

Industry Reviews

'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
'The book contains many examples and exercises which help the reader understand and assimilate the material. Also bibliographical comments at the end of each chapter provide useful references for further reading.' Bulletin of the American Mathematical Society

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Preface
Introduction
Malliavin operators in the one-dimensional case
Malliavin operators and isonormal Gaussian processes
Stein's method for one-dimensional normal approximations
Multidimensional Stein's method
Stein meets Malliavin: univariate normal approximations
Multivariate normal approximations
Exploring the Breuer-Major Theorem
Computation of cumulants
Exact asymptotics and optimal rates
Density estimates
Homogeneous sums and universality
Gaussian elements, cumulants and Edgeworth expansions
Hilbert space notation
Distances between probability measures
Fractional Brownian motion
Some results from functional analysis
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

Normal Approximations with Malliavin Calculus

From Stein's Method to Universality

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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