# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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## A novel series expansion for the multivariate normal probability integrals based on Fourier seriesHTML articles powered by AMS MathViewer

by Hatem A. Fayed and Amir F. Atiya
Math. Comp. 83 (2014), 2385-2402 Request permission

## Abstract:

In this article, we derive a series expansion of the multivariate normal probability integrals based on Fourier series. The basic idea is to transform the limits of each integral from $h_i$ to $\infty$ to be from $-\infty$ to $\infty$ by multiplying the integrand by a periodic square wave that approximates the domain of the integral. This square wave is expressed by its Fourier series expansion. Then a Cholesky decomposition of the covariance matrix is applied to transform the integrand to a simple one that can be easily evaluated. The resultant formula has a simple pattern that is expressed as multiple series expansion of trigonometric and exponential functions.
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• Hatem A. Fayed
• Affiliation: Department of Engineering, Mathematics and Physics, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
• Email: h{_}fayed@eng.cu.edu.eg
• Amir F. Atiya
• Affiliation: Department of Computer Engineering, Faculty of Engineering, Cairo University, Cairo, Egypt 12613
• Email: amir@alumni.caltech.edu
• Received by editor(s): June 28, 2012
• Received by editor(s) in revised form: January 9, 2013
• Published electronically: April 17, 2014
• © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: Math. Comp. 83 (2014), 2385-2402
• MSC (2010): Primary 42A16, 62H86
• DOI: https://doi.org/10.1090/S0025-5718-2014-02844-5
• MathSciNet review: 3223336