Fourth moment theorems on the Poisson space in any dimension

Abstract

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.

arXiv commentMinor revision. to appear in Electron. J. Probab

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Cited by in corpus (7)

• A simplified second-order Gaussian Poincaré inequality in discrete setting with applications
• Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation
• The multivariate functional de Jong CLT
• Malliavin-Stein Method: a Survey of Recent Developments
• The fourth moment theorem on the Poisson space
• Stable limit theorems on the Poisson space
• Fourth moment theorems on the Poisson space: analytic statements via product formulae