  We characterize the asymptotic independence between blocks consisting of
multiple Wiener-It\^{o} integrals. As a consequence of this characterization,
we derive the celebrated fourth moment theorem of Nualart and Peccati, its
multidimensional extension and other related results on the multivariate
convergence of multiple Wiener-It\^{o} integrals, that involve Gaussian and non
Gaussian limits. We give applications to the study of the asymptotic behavior
of functions of short and long-range dependent stationary Gaussian time series
and establish the asymptotic independence for discrete non-Gaussian chaoses.
