Characterizations of the Laplace and related distributions via geometric compound

Let Fa,\ be the distribution with Linnik's characteristic fonction {t) = 1/(1+? | ?|?), where A >0 axid a (0, 2]. In this paper we first investigate the basic properties of Fa,\, including the self-decomposability and the existence of moments. Then using an elementary approach we prove two characterization theorems for Fai\, which involve the stability of the geometric compound of a sequence of i.i.d. random variables.