Malliavin-Stein Method: a Survey of Recent Developments

Abstract

Initiated around the year 2007, the Malliavin-Stein approach to probabilistic approximations combines Stein's method with infinite-dimensional integration by parts formulae based on the use of Malliavin-type operators. In the last decade, Malliavin-Stein techniques have allowed researchers to establish new quantitative limit theorems in a variety of domains of theoretical and applied stochastic analysis. The aim of this survey is to illustrate some of the latest developments of the Malliavin-Stein method, with specific emphasis on extensions and generalisations in the framework of Markov semigroups and of random point measures.

arXiv admin note: text overlap with arXiv:1009.1310 by other authors

References in corpus (11)

• Central limit theorems for sequences of multiple stochastic integrals
• Noncentral convergence of multiple integrals
• Chaos of a Markov operator and the fourth moment condition
• Stein kernels and moment maps
• Stein's method for normal approximation in Wasserstein distances with application to the multivariate Central Limit Theorem
• A bound on the 2-Wasserstein distance between linear combinations of independent random variables
• Fourth moment theorems on the Poisson space in any dimension
• Stein's method for multivariate Brownian approximations of sums under dependence
• Donsker's theorem in {Wasserstein}-1 distance
• New moments criteria for convergence towards normal product/tetilla laws
• Existence of Stein Kernels under a Spectral Gap, and Discrepancy Bound