Multivariate Generalized Gram-Charlier Series in Vector Notations

TL;DR

This paper introduces a vector notation approach to the multivariate Generalized Gram-Charlier series, simplifying higher-order derivatives using Kronecker products, making the series more elementary and accessible.

Contribution

It derives the multivariate GGC series in vector notation, avoiding tensor calculus and simplifying the computation of derivatives for joint PDFs.

Findings

Vector notation simplifies multivariate derivatives

Provides elementary calculus-based derivations

Enhances the understanding of multivariate cumulants and moments

Abstract

The article derives multivariate Generalized Gram-Charlier (GGC) series that expands an unknown joint probability density function (\textit{pdf}) of a random vector in terms of the differentiations of the joint \textit{pdf} of a reference random vector. Conventionally, the higher order differentiations of a multivariate \textit{pdf} in GGC series will require multi-element array or tensor representations. But, the current article derives the GGC series in vector notations. The required higher order differentiations of a multivariate \textit{pdf} in vector notations are achieved through application of a specific Kronecker product based differentiation operator. Overall, the article uses only elementary calculus of several variables; instead Tensor calculus; to achieve the extension of an existing specific derivation for GGC series in univariate to multivariate. The derived multivariate GGC expression is more elementary as using vector notations compare to the coordinatewise tensor notations and more comprehensive as apparently more nearer to its counterpart for univariate. The same advantages are shared by the other expressions obtained in the article; such as the mutual relations between cumulants and moments of a random vector, integral form of a multivariate \textit{pdf}, integral form of the multivariate Hermite polynomials, the multivariate Gram-Charlier A (GCA) series and others.