Asymptotic expansions relating to the distribution of the product of correlated normal random variables

TL;DR

This paper derives asymptotic expansions for the tail distribution and quantile function of the product of correlated normal variables, providing tools for better approximation in statistical applications involving such products.

Contribution

It introduces new asymptotic expansions for the distribution and quantiles of products of correlated normal variables, extending previous results to more general cases.

Findings

Asymptotic approximations closely match numerical results

Effective for tail probability estimation

Applicable to sums of independent copies of the variables

Abstract

Asymptotic expansions are derived for the tail distribution of the product of two correlated normal random variables with non-zero means and arbitrary variances, and more generally the sum of independent copies of such random variables. Asymptotic approximations are also given for the quantile function. Numerical results are given to test the performance of the asymptotic approximations.