Expansions for multivariate densities

•This paper developed the multivariate Gram–Charlier series by Woodroofe–Stein’s identity.•This paper proposed a modified series for better approximation property.

The Gram–Charlier and Edgeworth series are expansions of probability distribution in terms of its cumulants. The expansions for the multivariate case have not been fully explored. This paper aims to develop the multivariate Gram–Charlier series by Woodroofe–Stein’s identity, and improve its approximation property by using the scaled normal density and Hermite polynomials. The series are useful to reconstruct the probability distribution from measurable higher moments.