Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses

Abstract

In this paper, we establish Poisson limit theorems on Poisson and Rademacher chaoses. Our principal result is a total-variation bound, valid in both settings, for an integer-valued functional whose highest-order chaos is dominant. The approximation error is the sum of the pure-chaos four-moment terms and an explicit remainder controlled by the $L^4$-size of the lower-order chaoses. On Poisson space the pure-chaos term is controlled solely by the moment defect, whereas on Rademacher space an additional maximal-influence correction is required. When the lower-order remainder vanishes, we recover exactly the corresponding total-variation bound for a shifted pure chaos. As consequences, for nonnegative integer-valued shifts of random variables in a fixed Poisson chaos, convergence of the first four moments is equivalent to convergence in total variation to a Poisson law, together with uniform integrability of the fourth powers; on a fixed Rademacher chaos, the analogous conclusion holds under a vanishing maximal-influence condition. The proof reveals a unit-jump rigidity phenomenon: the four-moment defect simultaneously suppresses unwanted spectral components and rules out non-unit jumps. We show, through an explicit quadratic counterexample, that the maximal-influence condition in the Rademacher setting is necessary for a general Poisson limit theorem. Finally, for every order $q\geq2$, we construct pure Poisson-chaos sequences with vanishing moment defect that converge weakly to a centered Poisson law. These examples show that the exact higher-order rigidity is not uniform once the lattice condition is removed. In both the Poisson and Rademacher settings, our proofs follow a unified strategy combining the Chen--Stein method, exchangeable pairs, and Ledoux's spectral viewpoint.