Analytic and Asymptotic Properties of Multivariate Generalized Linnik’s Probability Densities

Abstract

This paper studies the properties of the probability density function p α,ν,n(x) of the n-variate generalized Linnik distribution whose characteristic function φ α,ν,n(t) is given by $$\varphi_{\alpha,\nu,n}(\boldsymbol{t})=\frac{1}{(1+\Vert\boldsymbol{t}\Vert^{\alpha})^{\nu}},\quad\alpha\in (0,2],\ \nu>0,\ \boldsymbol{t}\in\mathbb{R}^n,$$ where ‖t‖ is the Euclidean norm of t∈ℝn. Integral representations of p α,ν,n(x) are obtained and used to derive the asymptotic expansions of p α,ν,n(x) when ‖x‖→0 and ‖x‖→∞ respectively. It is shown that under certain conditions which are arithmetic in nature, p α,ν,n(x) can be represented in terms of entire functions.