Fourth-Moment Theorems for Sums of Multiple Integrals

Nualart & Pecatti ([Nualart and Peccati, 2005, Thm 1]) established the first fourth-moment theorem for random variables in a fixed Wiener chaos, i.e. they showed that convergence of the sequence of fourth moments to the fourth moment of the standard Gaussian distribution is sufficient for weak convergence to the standard Gaussian. In this paper, we provide what we believe to be the first generalization to chaos expansions with more than a single term. Specifically, we show that a fourth-moment theorem holds for random variables consisting of sums of two multiple integrals of orders $p, q \in N$, where $p, q$ have different parities. Furthermore, we show that such random variables cannot themselves be Gaussian, again generalizing what is known for the fixed Wiener chaos setting. Finally, we show a fourth-moment theorem for variables with infinite Wiener chaos expansions when the terms in the expansions are independent and satisfy an additional regularity condition in terms of the Ornstein-Uhlenbeck operator.