Orthogonal polynomials generated by random vectors

Christopher S. Withers and Saralees Nadarajah

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 12, 6130-6136

Abstract: Every random q-vector with finite moments generates a set of orthonormal polynomials. These are generated from the basis functions xn = xn11…xnqq using Gram–Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.

Date: 2017
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DOI: 10.1080/03610926.2015.1118512

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