Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

Abstract

We develop the asymptotic expansion theory for vector-valued sequences (FN)N≥1 of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence FN. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of FN and we illustrate our results by several examples.