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Normal Approximations with Malliavin Calculus
Ivan Nourdin,Giovanni Peccati +1 more
TLDR
In this article, the authors provide an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space, and explain the connections between Stein's methods and Mallian calculus of variations.Abstract:
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.read more
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Stein's method for comparison of univariate distributions
TL;DR: In this article, a new general version of Stein's method for univariate distributions is proposed, which is based on a linear difference or differential-type operator, and the resulting Stein identity highlights the unifying theme behind the literature on Stein's methods both for continuous and discrete distributions.
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Stein's method, logarithmic Sobolev and transport inequalities
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Convergence in total variation on Wiener chaos
Ivan Nourdin,Guillaume Poly +1 more
TL;DR: In this article, it was shown that the Peccati-tudor theorem holds in the total variation topology of a sequence of random variables belonging to a finite sum of non-Gaussian chaoses.