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.
"Fourth moment theorems on the Poisson space: analytic statements via product formulae." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP196