Strong asymptotic independence on Wiener chaos

Let F_n = (F_{1,n}, \dots ,F_{d,n}), be a sequence of random vectors such that, for every j=1,\dots ,d belongs to a fixed Wiener chaos of a Gaussian field. We show that, as n\to \infty are asymptotically independent if and only if \textup {Cov}(F_{i,n}^2,F_{j,n}^2)\to 0. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosiński (2014).