Malliavin–Stein method: a survey of some recent developments

Azmoodeh, Ehsan; Peccati, Giovanni; Yang, Xiaochuan

doi:10.15559/21-VMSTA184

Modern Stochastics: Theory and Applications

Malliavin–Stein method: a survey of some recent developments

Volume 8, Issue 2 (2021), pp. 141–177

Ehsan Azmoodeh Giovanni Peccati Xiaochuan Yang

https://doi.org/10.15559/21-VMSTA184

Pub. online: 22 June 2021 Type: Research Article

Open Access

Received
12 February 2021

Revised
28 May 2021

Accepted
10 June 2021

Published
22 June 2021

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 generalizations in the framework of Markov semigroups and of random point measures.

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Keywords

Limit theorems Stein’s method Malliavin calculus Wiener space Poisson space multiple integral Markov triple Markov generator eigenspace eigenfunction spectrum functional Γ-calculus weak convergence fourth moment theorems Berry–Essen bounds probability metrics 60F05 60B10 28C20 60H07 47D07 34L10 47A10

Funding

Giovanni Peccati is supported by the FNR grant FoRGES (GS1R-AGR-3376-10) at Luxembourg University. Xiaochuan Yang is supported by the EPSRC grant EP/T028653/1.

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