A novel series expansion for the multivariate normal probability integrals based on Fourier series

Fayed, Hatem; Atiya, Amir

doi:10.1090/S0025-5718-2014-02844-5

A novel series expansion for the multivariate normal probability integrals based on Fourier series
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by Hatem A. Fayed and Amir F. Atiya;

Math. Comp. 83 (2014), 2385-2402

DOI: https://doi.org/10.1090/S0025-5718-2014-02844-5

Published electronically: April 17, 2014

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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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