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The fourth moment theorem on the Poisson space. (English) Zbl 1431.60022

Summary: We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result – that has been elusive for several years – shows that the so-called ‘fourth moment phenomenon’, first discovered by D. Nualart and the second author [Ann. Probab. 33, No. 1, 177–193 (2005; Zbl 1097.60007)] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein’s method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by M. Ledoux [Ann. Probab. 40, No. 6, 2439–2459 (2012; Zbl 1266.60042)] and E. Azmoodeh et al. [J. Funct. Anal. 266, No. 4, 2341–2359 (2014; Zbl 1292.60078)]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

MSC:

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