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

TL;DR

This work studies the distribution of the product $Z=XY$ of correlated normals and the sum $S_n=sum_{i=1}^n Z_i$ when $(X,Y)$ has nonzero means and arbitrary variances. It derives the full asymptotic expansion for the PDF of $S_n$ as $|x| oty$, and uses this to obtain tail probabilities and quantile approximations, with special-case simplifications when $r_Xr_Y=0$ or $r_X+r_Y=0$. The expansions are expressed in terms of confluent hypergeometric functions $M$ and $U$ and Puiseux-series coefficients $g_{i,j}(a,b)$, enabling fast, accurate computations. Numerical experiments validate the accuracy across regimes and demonstrate the practical usefulness for tail-risk assessment and quantile estimation in statistical and physical applications.

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.

Theorems & Definitions (11)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Theorem 3.5
• Remark 3.6
• Theorem 3.7
• Remark 3.8
• Lemma 5.1
• Lemma 5.2