Electronic Communications in Probability

Fourth moment theorems on the Poisson space: analytic statements via product formulae

Christian Döbler and Giovanni Peccati

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We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carré-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Döbler and Peccati (2018) and Döbler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 91, 12 pp.

Received: 9 August 2018
Accepted: 16 November 2018
First available in Project Euclid: 15 December 2018

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Zentralblatt MATH identifier

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

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

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Döbler, Christian; Peccati, Giovanni. Fourth moment theorems on the Poisson space: analytic statements via product formulae. Electron. Commun. Probab. 23 (2018), paper no. 91, 12 pp. doi:10.1214/18-ECP196. https://projecteuclid.org/euclid.ecp/1544843114

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