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

¹ Applied Mathematics Group Industrial Research Limited Lower Hutt, New Zealand
² School of Mathematics University of Manchester Manchester M13 9PL, UK. mbbsssn2@manchester.ac.uk

Received: 19 July 2009

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 < Y < b) for Y ~ N_p(0,V) and for repeated integrals of the density of Y. When V⁻¹y > 0 in R³ the expansion for P(Y < 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.