Right-tail asymptotics for products of independent normal random variables

TL;DR

This paper derives explicit asymptotic formulas for the probability that the product of independent normal variables exceeds a large threshold, accounting for different sign patterns and providing correction terms.

Contribution

It provides a novel boundary saddle-point/Laplace method to approximate right tail probabilities for products of normal variables with explicit correction terms.

Findings

Asymptotic approximation includes a finite multiplicative factor for nonzero means.

Explicit first correction term of order x^{-1/n} is derived.

Remaining error is of order x^{-2/n}.

Abstract

Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(\mu_i,\sigma_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it includes an explicit first relative correction term of order $x^{-1/n}$, with remaining relative error $O(x^{-2/n})$. The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation.