Universal Gaussian fluctuations on the discrete Poisson chaos

Abstract

We prove that homogeneous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Solé, Taqqu and Utzet (Ann. Probab. 38 (2010) 443–478) and by Peccati and Zheng (Electron. J. Probab. 15 (2010) 1487–1527).