Gaussian approximation of Gaussian scale mixture

TL;DR

This paper investigates the optimal Gaussian approximation of Gaussian scale mixtures, both in scalar and multivariate cases, by minimizing the $L^2$ distance between the mixture and a Gaussian with a fixed variance.

Contribution

It provides a method to compute the best Gaussian approximation for Gaussian scale mixtures in scalar and multivariate settings.

Findings

Derived explicit formulas for the optimal scalar $t_0$.

Extended the approximation framework to multivariate Gaussian scale mixtures.

Demonstrated the effectiveness of the approximation in theoretical and practical scenarios.

Abstract

For a given positive random variable $V>0$ and a given $Z\sim N(0,1)$ independent of $V$, we compute the scalar $t_0$ such that the distance between $Z\sqrt{V}$ and $Z\sqrt{t_0}$ in the $L^2(\R)$ sense, is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.