On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension

TL;DR

This paper derives a stochastic representation and asymptotic distribution for the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions, facilitating faster simulations and understanding of their properties.

Contribution

It introduces a novel stochastic representation and asymptotic analysis for the product of singular Wishart matrices and Gaussian vectors, enhancing simulation efficiency and theoretical understanding.

Findings

Derived a stochastic representation for the product

Established the characteristic function and asymptotic distribution

Demonstrated good performance of the asymptotic distribution in simulations

Abstract

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.