The fourth moment theorem on the Poisson space

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 Nualart and Peccati [Ann. Probab. 33 (2005) 177.193] 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 Ledoux [Ann. Probab. 40 (2012) 2439.2459] and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341.2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.