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

Christopher S. Withers; Saralees Nadarajah

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 340-357
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Withers, Christopher S., and Nadarajah, Saralees. "Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal." ESAIM: Probability and Statistics 15 (2012): 340-357. <http://eudml.org/doc/222459>.

@article{Withers2012,
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. },
author = {Withers, Christopher S., Nadarajah, Saralees},
journal = {ESAIM: Probability and Statistics},
keywords = {Asymptotic expansion; Leibniz' rule; repeated integrals of products; multivariate Hermite polynomials; multivariate normal; asymptotic expansion; Leibniz rule; multivariate Hermite polynomials},
language = {eng},
month = {1},
pages = {340-357},
publisher = {EDP Sciences},
title = {Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal},
url = {http://eudml.org/doc/222459},
volume = {15},
year = {2012},
}

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
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 340
EP - 357
AB - 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.
LA - eng
KW - Asymptotic expansion; Leibniz' rule; repeated integrals of products; multivariate Hermite polynomials; multivariate normal; asymptotic expansion; Leibniz rule; multivariate Hermite polynomials
UR - http://eudml.org/doc/222459
ER -

References

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