On the distribution-tail behaviour of the product of normal random variables

Abstract

Abstract In this paper we consider the product Π n = ∏ k = 1 n ξ k $\Pi _{n}=\prod_{k=1}^{n}\xi _{k}$ of n independent normally distributed zero mean random variables ξ 1 , … , ξ n $\xi _{1},\dots ,\xi _{n}$ . We derive an asymptotic formula for the survival probability P ( Π n > x ) ${\mathbb{P}}(\Pi _{n}>x)$ , as x → ∞ $x\to \infty $ , with the first remaining term.

Keywords

• Distribution tail
• Product of random variables
• Normal distribution
• Saddle point method
• Asymptotic formulas