Multiplying a Gaussian Matrix by a Gaussian Vector

TL;DR

This paper characterizes the multivariate generalized Laplace distribution and shows that multiplying a Gaussian matrix with i.i.d. columns by an independent isotropic Gaussian vector results in a symmetric multivariate generalized Laplace distribution.

Contribution

It provides a new simple characterization of the multivariate generalized Laplace distribution and establishes a novel distributional result for Gaussian matrix-vector products.

Findings

Product of Gaussian matrix and isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution

New characterization simplifies understanding of the multivariate generalized Laplace distribution

Results have potential applications in multivariate statistical modeling

Abstract

We provide a new and simple characterization of the multivariate generalized Laplace distribution. In particular, this result implies that the product of a Gaussian matrix with independent and identically distributed columns by an independent isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution.