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.
Mots-clés : asymptotic expansion, Leibniz' rule, repeated integrals of products, multivariate Hermite polynomials, multivariate normal
@article{PS_2011__15__340_0,
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},
pages = {340--357},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2010005},
mrnumber = {2870519},
zbl = {1266.60024},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2010005/}
}
TY - JOUR AU - Withers, Christopher S. AU - Nadarajah, Saralees TI - Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal JO - ESAIM: Probability and Statistics PY - 2011 SP - 340 EP - 357 VL - 15 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2010005/ DO - 10.1051/ps/2010005 LA - en ID - PS_2011__15__340_0 ER -
%0 Journal Article %A Withers, Christopher S. %A Nadarajah, Saralees %T Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal %J ESAIM: Probability and Statistics %D 2011 %P 340-357 %V 15 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2010005/ %R 10.1051/ps/2010005 %G en %F PS_2011__15__340_0
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. http://www.numdam.org/articles/10.1051/ps/2010005/
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