Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 340-357

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.

DOI : 10.1051/ps/2010005
Classification : 60E05, 62H05
Keywords: asymptotic expansion, Leibniz' rule, repeated integrals of products, multivariate Hermite polynomials, multivariate normal 
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     author = {Withers, Christopher S. and Nadarajah, Saralees},
     title = {Expansions for {Repeated} {Integrals} of {Products} with {Applications} to the {Multivariate} {Normal}},
     journal = {ESAIM: Probability and Statistics},
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Withers, Christopher S.; Nadarajah, Saralees. Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 340-357. doi: 10.1051/ps/2010005

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