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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
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.
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- Research Article
- Information
- ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 340 - 357
- Copyright
- © EDP Sciences, SMAI, 2011