Annals of Probability

The fourth moment theorem on the Poisson space

Christian Döbler and Giovanni Peccati

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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.

Article information

Ann. Probab., Volume 46, Number 4 (2018), 1878-1916.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H05: Stochastic integrals

Poisson functionals multiple Wiener–Itô integrals fourth moment theorem carré-du-champ operator Berry–Esseen bounds Gaussian approximation Gamma approximation Malliavin calculus Stein’s method


Döbler, Christian; Peccati, Giovanni. The fourth moment theorem on the Poisson space. Ann. Probab. 46 (2018), no. 4, 1878--1916. doi:10.1214/17-AOP1215.

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