Infinite Divisibility of the Product of Two Correlated Normal Random Variables and Exact Distribution of the Sample Mean

TL;DR

This paper proves that the product of two correlated normal variables is infinitely divisible and derives exact formulas for the distribution of their sums, advancing understanding of their probabilistic properties.

Contribution

It introduces the first proof of infinite divisibility for the product of correlated normal variables and provides exact distribution formulas for sums of such variables.

Findings

Product of correlated normals is infinitely divisible.

Derived exact distribution formulas for sums of these variables.

Enhanced theoretical understanding of their probabilistic structure.

Abstract

We prove that the distribution of the product of two correlated normal random variables with arbitrary means and arbitrary variances is infinitely divisible. We also obtain exact formulas for the probability density function of the sum of independent copies of such random variables.