Mathematics – Probability
Scientific paper
2009-03-30
J. Fourier Anal. Appl. 16 (2010), 715-747
Mathematics
Probability
28 pages
Scientific paper
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.
Lim Chu S.
Teo Lee Peng
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