Right-tail asymptotics for products of independent normal random variables

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.