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

Published online by Cambridge University Press:  05 January 2012

Christopher S. Withers  and
Saralees Nadarajah
Christopher S. Withers
Affiliation:
Applied Mathematics Group Industrial Research Limited Lower Hutt, New Zealand
Saralees Nadarajah
Affiliation:
School of Mathematics University of Manchester Manchester M13 9PL, UK. mbbsssn2@manchester.ac.uk
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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 ~ 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.

Keywords

Asymptotic expansionLeibniz' rulerepeated integrals of productsmultivariate Hermite polynomialsmultivariate normal
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 340 - 357
Copyright
© EDP Sciences, SMAI, 2011

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