A mixture representation of the Linnik distribution

Cataloged from PDF version of article.Linnik distribution with the characteristic function ~o,(tl = 1/(1 + Itl=), 0 < ~ < 2, is shown to possess the following property. Let X,,Xp be random variables possessing the Linnik distribution with parameters ~ and (0 < ~ < fl ~< 2). Denote by Y~ an independent of X~ non-negative random variable with the density ,q(s;~,fl) = sin 1 + s 2~ + 2s cos~ Then X, -'- X e Y~p, fl respectively where - denotes the equality in the sense of distributions. Infinite divisibility of mixtures of Linnik distributions with respect to the parameter ~ and scale is obtained as a corollary. AMS 1980 Subject Classification." Primary 62H05, 60El0; Secondary 33A4

Analytic and asymptotic properties of generalized Linnik probability densities

Cataloged from PDF version of article.This paper studies the properties of the probability density function pα,ν,n(x) of the n-variate generalized Linnik distribution whose characteristic function φα,ν,n(t) is given by where {norm of matrix}t{norm of matrix} is the Euclidean norm of t ∈ ℝn. Integral representations of pα,ν,n(x) are obtained and used to derive the asymptotic expansions of pα,ν,n(x) when {norm of matrix}x{norm of matrix}→0 and {norm of matrix}x{norm of matrix}→∞ respectively. It is shown that under certain conditions which are arithmetic in nature, pα,ν,n(x) can be represented in terms of entire functions. © 2009 Birkhäuser Boston