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 $\varphi_{\alpha,\nu,n}(\boldsymbol{t})$ is given by \varphi_{\alpha,\nu,n}(\boldsymbol{t})=\frac{1} {(1+\Vert\boldsymbol{t}\Vert^{\alpha})^{\nu}}, \alpha\in (0,2], \nu>0, \boldsymbol{t}\in \mathbb{R}^n, where $\Vert\boldsymbol{t}\Vert$ is the Euclidean norm of $\boldsymbol{t}\in\mathbb{R}^n$. 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.