Strong asymptotic independence on Wiener chaos

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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(Fᵢ,ₙ²,Fⱼ,ₙ²)→ 0 for every i≠ 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 Rosinski.