## Electronic Communications in Probability

### Fourth moment theorems on the Poisson space: analytic statements via product formulae

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 91, 12 pp.

Dates
Accepted: 16 November 2018
First available in Project Euclid: 15 December 2018

https://projecteuclid.org/euclid.ecp/1544843114

Digital Object Identifier
doi:10.1214/18-ECP196

Mathematical Reviews number (MathSciNet)
MR3896829

Zentralblatt MATH identifier
07023477

#### Citation

Döbler, Christian; Peccati, Giovanni. Fourth moment theorems on the Poisson space: analytic statements via product formulae. Electron. Commun. Probab. 23 (2018), paper no. 91, 12 pp. doi:10.1214/18-ECP196. https://projecteuclid.org/euclid.ecp/1544843114

#### References

• [3] C. Döbler and G. Peccati, Limit theorems for symmetric $U$-statistics using contractions, arXiv:1802.00394 (2018).