Stable Densities Under Change of Scale and Total Variation Inequalities

Abstract

In this paper it is shown that if $q$ is the density of a symmetric stable density, then for $c \in (0, 1) \cup (1, \infty)$, the graph of $q(x)$ intersects the graph of $cq(cx)$ at only two points. The argument proceeds by introducing a new characterization of unimodality for densities and involves a representation for symmetric stable random variables that is also useful for simulating such random variables. Finally our results are applied to prove some inequalities concerning the total variation norm of the difference of two symmetric stable densities.