Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal

Abstract

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for P (a ) for Y ~ Np (0,V ) and for repeated integrals of the density of Y . When V −1 y > 0 in R 3 the expansion for P (Y ) reduces to one given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of N p (0,V −1 ). This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials. These are given explicitly.