Non-symmetric Linnik distributions

Abstract

The aim of this Note is to study the probability density with characteristic function ρα,θ,v(t) = 1/(1 + e-iθsgnt|t|α)v, where 0 < α < 2, |θ| ≤ min(πα/2, π - πα/2), and v > 0. This density, first introduced by Linnik for θ = 0, v = 1, received several applications later. It does not have any explicit representation. We consider here its integral and series representations and its analytical properties.