Strong asymptotic independence on Wiener chaos

Let Fn = (F1,n, ...., Fd,n), n > 1, be a sequence of random vectors such that, for every j = 1, ..., d, the random variable Fj,n belongs to a fixed Wiener chaos of a Gaussian field. We show that, as n → ∞, the components of Fn are asymptotically independent if and only if Cov(F 2 i,n, F 2 j,n) → 0 for every i 6= j. 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 [9].