Normal approximations with Malliavin calculus : from Stein's method to universality / Ivan Nourdin, Giovanni Peccati.

Format:

Book

Author/Creator:

Nourdin, Ivan, author.

Peccati, Giovanni, 1975- author.

Series:

Cambridge tracts in mathematics ; 192.

Cambridge tracts in mathematics ; 192

Language:

English

Subjects (All):

Approximation theory.

Malliavin calculus.

Physical Description:

1 online resource (xiv, 239 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2012.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-107-23077-2

1-280-87800-2

1-139-37878-3

9786613719317

1-139-37592-X

1-139-08465-8

1-139-38021-4

1-139-37193-2

1-139-37735-3

OCLC:

797919775