A note on the distribution of the product of zero mean correlated normal random variables

TL;DR

This paper simplifies the derivation of the probability density function for the product of two zero mean correlated normal variables by identifying it as a variance-gamma distribution, resolving a long-standing problem.

Contribution

It provides a simple proof that the product follows a variance-gamma distribution, offering an explicit formula for its probability density function.

Findings

Identifies the product as a variance-gamma distribution

Provides an explicit formula for the probability density function

Simplifies the proof of the distribution's form

Abstract

The problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables dates back to 1936. Perhaps surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero mean correlated normal random variables as a variance-gamma random variable, from which an explicit formula for probability density function is immediate.