How Broad is the Class of Normal Scale Mixtures?

Abstract

We study the class of scale mixtures of normal distributions with mean zero. Given that the $\operatorname{cdf} F(x)$ of such a mixture is fixed at two points, say $F(x_1) = \alpha_1, F(x_2) = \alpha_2$, we answer the question of how widely $F(x_3)$ can vary at some third point $x_3$. A brief final section mentions extensions of our theorem.