Fourth-moment theorems for sums of multiple integrals

In this paper, we extend the celebrated fourth-moment theorem of Nualart and Peccati, which states that convergence of the fourth moment to 3 3 implies weak convergence to the standard Gaussian for random variables with zero mean and unit variance in a fixed Wiener chaos. Our main result establishes that the same conclusion holds for random variables expressible as sums of two components in the p p -th and q q -th Wiener chaoses, provided p p and q q have different parities. We also prove quantitative bounds in the 1 1 -Wasserstein and total variation distances, both scaling with the square root of the fourth cumulant. To the best of our knowledge, this is the first fourth-moment theorem that applies to nontrivial sums across different chaos levels. Finally, we show via a counterexample with p = 1 p = 1 , q = 3 q = 3 that the result fails when p p and q q share the same parity.