## 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 PDF
- 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.## References

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## Additional Information

**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