Electronic Journal of Probability

Fourth moment theorems on the Poisson space in any dimension

Christian Döbler, Anna Vidotto, and Guangqu Zheng

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We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.

Finally, a transfer principle “from-Poisson-to-Gaussian” is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 36, 27 pp.

Received: 28 September 2017
Accepted: 13 April 2018
First available in Project Euclid: 3 May 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

multivariate Poisson functionals multiple Wiener-Itô integrals fourth moment theorems carré du champ operator Gaussian approximation Stein’s method exchangeable pairs Peccati-Tudor theorem

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Döbler, Christian; Vidotto, Anna; Zheng, Guangqu. Fourth moment theorems on the Poisson space in any dimension. Electron. J. Probab. 23 (2018), paper no. 36, 27 pp. doi:10.1214/18-EJP168. https://projecteuclid.org/euclid.ejp/1525312960

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