Orthogonal polynomials generated by random vectors

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.

KEYWORDS:

• Hermite polynomial
• multivariate normal distribution
• orthogonal polynomials.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62E99

Acknowledgments

The authors would like to thank the Editor, the Associate Editor, and the two referees for careful reading and comments which greatly improved the paper.