A mixture representation of the Linnik distribution

Linnik distribution with the characteristic function [phi][alpha](t)=1/(1+t[alpha]), 0<[alpha]<2, is shown to possess the following property. Let X[alpha],X[beta] be random variables possessing the Linnik distribution with parameters [alpha] and [beta] respectively (0 < [alpha] < [beta] [less-than-or-equals, slant] 2). Denote by Y[alpha][beta] an independent of X[beta] non-negative random variable with the density , 0<s<infin;. Then X[alpha][circle, equals]X[beta]Y[alpha][beta] where RS denotes the equality in the sense of distributions. Infinite divisibility of mixtures of Linnik distributions with respect to the parameter [alpha] and scale is obtained as a corollary.