Fourth moment theorems on the Poisson space: analytic statements via product formulae. (English) Zbl 1406.60037

Döbler, Christian; Peccati, Giovanni

Electron. Commun. Probab. 23, Paper No. 91, 12 p. (2018).

Summary: 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 C. Döbler et al. [Ann. Probab. 46, No. 4, 1878–1916 (2018; Zbl 1431.60022); Electron. J. Probab. 23, Paper No. 36, 27 p. (2018; Zbl 1387.60045)]. An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

MSC:

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

Keywords:

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

Citations:

Zbl 1431.60022; Zbl 1387.60045

References:

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