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

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.


60F05 Central limit and other weak theorems
60H07 Stochastic calculus of variations and the Malliavin calculus
60H05 Stochastic integrals
Full Text: DOI arXiv Euclid


[1] https://sites.google.com/site/malliavinstein/home.
[2] C. Döbler and G. Peccati, Quantitative de jong theorems in any dimension, Electronic Journal of Probability 22 (2016), Paper 35.
[3] C. Döbler and G. Peccati, Limit theorems for symmetric \(U\)-statistics using contractions, arXiv:1802.00394 (2018).
[4] C. Döbler and G. Peccati, The fourth moment theorem on the Poisson space, Ann. Probab. 46 (2018), no. 4, 1878–1916. · Zbl 1431.60022
[5] C. Döbler, A. Vidotto, and G. Zheng, Fourth moment theorems on the Poisson space in any dimension, Electron. J. Probab. 23 (2018), 1–27. · Zbl 1387.60045
[6] G. Last, Stochastic analysis for Poisson processes, Stochastic analysis for Poisson point processes (G. Peccati and M. Reitzner, eds.), Mathematics, Statistics, Finance and Economics, Bocconi University Press and Springer, 2016, pp. 1–36.
[7] G. Last, G. Peccati, and M. Schulte, Normal approximation on poisson spaces: Mehler’s formula, second order poincaré inequalities and stabilization, Probab. Theory Related Fields (2016). · Zbl 1347.60012
[8] G. Last and M. Penrose, Lectures on the poisson process, IMS Textbooks, Cambridge University Press, Cambridge, 2017.
[9] G. Last and M. D. Penrose, Poisson process Fock space representation, chaos expansion and covariance inequalities, Probab. Theory Related Fields 150 (2011), no. 3-4, 663–690. · Zbl 1233.60026
[10] I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012, From Stein’s method to universality. · Zbl 1266.60001
[11] N. Privault, Stochastic analysis in discrete and continuous settings, Springer Berlin Heidelberg, 2009 (eng).
[12] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984), no. 2, 217–239. · Zbl 0548.60058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.