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



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Nourdin, Ivan.
Publication date:
Cambridge [England] ; New York : Cambridge University Press, 2012.
  • Book
  • xiv, 239 p. : ill ; 23 cm.
Includes bibliographical references (p. 227-234) and index.
  • 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.
"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"-- Provided by publisher. "This is a text about probabilistic approximations, which are mathematical statements providing estimates of the distance between the laws of two random objects. As the title suggests, we will be mainly interested in approximations involving one or more normal (equivalently called Gaussian) random elements. Normal approximations are naturally connected with central limit theorems (CLTs), i.e. convergence results displaying a Gaussian limit, and are one of the leading themes of the whole theory of probability"-- Provided by publisher.
Peccati, Giovanni, 1975-
Cambridge tracts in mathematics ; 192.

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