Analytic and asymptotic properties of multivariate generalized Linnik's probability densities

Abstract

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

28 pages

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