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Gaussian Systems, T-Hermite Polynomials, Moments, and Cumulants

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Multivariate Statistical Methods

Part of the book series: Frontiers in Probability and the Statistical Sciences ((FROPROSTAS))

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Abstract

Hermite polynomials have several applications in many fields of science. We start with the classical Hermite polynomials of one variable, which constitute a complete set of orthonormal polynomials in the nonlinear Hilbert space of Gaussian variates. We use the method of generating functions for deriving multivariate Hermite polynomials. Well known properties are listed and higher-order moments and cumulants are considered in detail not only for Hermite polynomials but also for Gaussian systems. These general formulae are given in connection with set partitions. Clear, computationally simple expressions are given for the product of two, three, and four Hermite polynomials in terms of linear combinations of Hermite polynomials. We use our T-calculus to study multivariate vector-valued Hermite polynomials. Most of the results for multivariate Hermite polynomials (scalar-valued) are generalized to vector-valued cases with the help of commutators, T-moments, and T-cumulants. We also establish relations between multivariate Hermite polynomials and multiple vector-valued Hermite polynomials. The Gram–Charlier expansion of multivariate distributions in terms of T-Hermite polynomials closes the chapter.

We consider the so-called probabilists’ Hermite polynomials, which physicists denote differently as He.

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Authors and Affiliations

  1. Faculty of Informatics, Dept. of IT, University of Debrecen, Debrecen, Hungary

    György Terdik

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  1. György Terdik
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Terdik, G. (2021). Gaussian Systems, T-Hermite Polynomials, Moments, and Cumulants. In: Multivariate Statistical Methods . Frontiers in Probability and the Statistical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-030-81392-5_4

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