On the product of correlated normal random variables and the noncentral chi-square difference distribution

TL;DR

This paper introduces a new distribution called the noncentral chi-square difference distribution, derived from correlated normal variables, providing exact formulas, characterizations, and properties for this distribution.

Contribution

It represents the product of correlated normals as a difference of noncentral chi-square variables and derives various analytical properties and formulas for this new distribution.

Findings

Exact formula for the probability density function.

Stein characterization of the distribution.

Self-decomposability and Lévy-Khintchine representation.

Abstract

We represent the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, as a difference of two independent noncentral chi-square random variables (which we refer to as the noncentral chi-square difference distribution). As a consequence, we obtain, amongst other results, an exact formula for the probability density function of the noncentral chi-square difference distribution, a Stein characterisation of the noncentral chi-square difference distribution, a simple formula for the moments of the sum of independent copies of the product of correlated normal random variables, an exact formula for the probability that such a random variable is negative, and also show that such random variables are self-decomposable and provide a L\'evy-Khintchine representation of the characteristic function.