Double $q$-Wigner Chaos and the Fourth Moment

In this paper, we prove the Fourth Moment Theorem for sequences of (noncommutative) random variables given as sums of two stochastic integrals in two different parity orders of chaos, both in the free Wigner chaos setting and a $q$-Gaussian generalization. Specifically, we prove that convergence to the appropriate central limit distribution is mediated entirely by the behavior of the first four (mixed) moments of the two stochastic integrals, which in turn controls the $L^2$ norms of partial integral contractions of those kernels. The key step in both the free and $q$-Gaussian settings is a polarization identity for fourth cumulants of sums which holds only when the two terms have differing parities. These results are analogous to the recent preprint Fourth-Moment Theorems for Sums of Multiple Integrals by Basse-O'Connor, Kramer-Bang, and Svedsen in the classical Wiener-It\^o chaos setting.