  We develop the asymptotic expansion theory for vector-valued sequences (F N)
N $\ge$1 of random variables in terms of the convergence of the Stein-Malliavin
matrix associated to the sequence F N. Our approach combines the classical
Fourier approach and the recent theory on Stein method and Malliavin calculus.
We find the second order term of the asymptotic expansion of the density of F N
and we illustrate our results by several examples. 2010 AMS Classification
Numbers: 62M09, 60F05, 62H12
