Scale Mixing of Symmetric Distributions with Zero Means

Abstract

Suppose that a distribution $A$ is a mixture of distributions similar to $B$ but with different scale parameters; or (almost equivalently) that a distribution $F$ is a convolution of a given distribution $G$ with some other distribution. We derive conditions on (i) the moments of $A$ and $F$ and (ii) on the derivatives of $A$ and $F$; these conditions are necessary, but are not sufficient in general. The conditions (ii) are appropriate when $B$ (or $G$) is of Polya type 3.