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July 2018 The fourth moment theorem on the Poisson space
Christian Döbler, Giovanni Peccati
Ann. Probab. 46(4): 1878-1916 (July 2018). DOI: 10.1214/17-AOP1215

Abstract

We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result—that has been elusive for several years—shows that the so-called ‘fourth moment phenomenon’, first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177–193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439–2459] and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341–2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

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Christian Döbler. Giovanni Peccati. "The fourth moment theorem on the Poisson space." Ann. Probab. 46 (4) 1878 - 1916, July 2018. https://doi.org/10.1214/17-AOP1215

Information

Received: 1 January 2017; Revised: 1 July 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 06919014
MathSciNet: MR3813981
Digital Object Identifier: 10.1214/17-AOP1215

Subjects:
Primary: 60F05
Secondary: 60H05 , 60H07

Keywords: Berry–Esseen bounds , carré-du-champ operator , Fourth moment theorem , gamma approximation , Gaussian approximation , Malliavin calculus , multiple Wiener–Itô integrals , Poisson functionals , Stein’s method

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 4 • July 2018
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