# 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

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topWithers, 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 -

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