On the infinite divisibility of scale mixtures of symmetric α-stable distributions, α ∈ (0,1]
The paper contains a new and elementary proof of the fact that if α ∈ (0,1] then every scale mixture of a symmetric α-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 α = 2. The problem discussed in the paper is still open for α ∈ (1,2).