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.

Minor revision. to appear in Electron. J. Probab

References in corpus (9)

• Central limit theorems for sequences of multiple stochastic integrals
• Exponential Approximation by Stein's Method and Spectral Graph Theory
• Chaos of a Markov operator and the fourth moment condition
• A note on the exchangeability condition in Stein's method
• Stein's method for dependent random variables occurring in Statistical Mechanics
• Exchangeable pairs on Wiener chaos
• On the fourth moment condition for Rademacher chaos
• The universality of homogeneous polynomial forms and critical limits
• The fourth moment theorem on the Poisson space

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• 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