Volume 15, 2011
|Page(s)||340 - 357|
|Published online||05 January 2012|
Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal
Applied Mathematics Group
Industrial Research Limited
Lower Hutt, New Zealand
2 School of Mathematics University of Manchester Manchester M13 9PL, UK. firstname.lastname@example.org
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 ~ Np(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 Np(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.
Mathematics Subject Classification: 60E05 / 62H05
Key words: Asymptotic expansion / Leibniz' rule / repeated integrals of products / multivariate Hermite polynomials / multivariate normal
© EDP Sciences, SMAI, 2011
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