The basic distributional theory for the product of zero mean correlated normal random variables

TL;DR

This paper reviews the fundamental distributional properties of the product of two zero mean correlated normal variables and their sums, highlighting their theoretical significance and applications in statistical modeling.

Contribution

It provides a comprehensive overview of the distributional theory for sums of such products, including derivations of key properties and connections to limiting distributions.

Findings

Derived probability and distribution functions

Identified representations in terms of other variables

Connected to limiting distribution in Wiener-Itô integrals

Abstract

The product of two zero mean correlated normal random variables, and more generally the sum of independent copies of such random variables, has received much attention in the statistics literature and appears in many application areas. However, many important distributional properties are yet to be recorded. This review paper fills this gap by providing the basic distributional theory for the sum of independent copies of the product of two zero mean correlated normal random variables. Properties covered include probability and cumulative distribution functions, generating functions, moments and cumulants, mode and median, Stein characterisations, representations in terms of other random variables, and a list of related distributions. We also review how the product of two zero mean correlated normal random variables arises naturally as a limiting distribution, with an example given for the distributional approximation of double Wiener-It\^{o} integrals.