On the infinite divisibility of scale mixtures of symmetric $\alpha$-stable distributions, $\alpha \in (0,1]$
The paper contains a new and elementary proof of the fact that if $\alpha \in (0,1]$ then every scale mixture of a symmetric $\alpha$-stable probability measure is infinitely divisible. This property is known to be a consequence of Kelker's result for the Cauchy distribution and some nontrivial properties of completely monotone functions. It is known that this property does not hold for $\alpha =2$. The problem discussed in the paper is still open for $\alpha \in (1,2)$.