A generalized Isserlis theorem for location mixtures of Gaussian random vectors

TL;DR

This paper extends the Isserlis theorem to general location mixtures of Gaussian vectors, including scale mixtures like the generalized hyperbolic distribution, broadening its applicability in probabilistic modeling.

Contribution

It generalizes previous work by Michalowicz et al. to all location mixtures of Gaussian vectors and provides an example for scale mixtures such as the generalized hyperbolic distribution.

Findings

Extended Isserlis theorem to all location mixtures of Gaussian vectors

Provided an example for scale mixtures like the generalized hyperbolic distribution

Broadens the theoretical tools available for Gaussian mixture models

Abstract

In a recent paper, Michalowicz et al. provide an extension of Isserlis theorem to the case of a Bernoulli location mixture of a Gaussian vector. We extend here this result to the case of any location mixture of Gaussian vector; we also provide an example of the Isserlis theorem for a "scale location" mixture of Gaussian, namely the d-dimensional generalized hyperbolic distribution.