Noncentral convergence of multiple integrals

Abstract

Fix ν>0, denote by G(ν/2) a Gamma random variable with parameter ν/2 and let n≥2 be a fixed even integer. Consider a sequence {F_k}_k≥1 of square integrable random variables belonging to the nth Wiener chaos of a given Gaussian process and with variance converging to 2ν. As k→∞, we prove that F_k converges in distribution to 2G(ν/2)−ν if and only if E(F_k⁴)−12E(F_k³)→12ν²−48ν.