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

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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.

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Authors and Affiliations

  1. Faculty of Engineering, Multimedia University, Jalan Multimedia, 63100, Cyberjaya, Selangor, Malaysia

    S. C. Lim

  2. Faculty of Information Technology, Multimedia University, Jalan Multimedia, 63100, Cyberjaya, Selangor, Malaysia

    L. P. Teo

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  1. S. C. Lim
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Correspondence to L. P. Teo.

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Communicated by Christian Houdré.

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Lim, S.C., Teo, L.P. Analytic and Asymptotic Properties of Multivariate Generalized Linnik’s Probability Densities. J Fourier Anal Appl 16, 715–747 (2010). https://doi.org/10.1007/s00041-009-9097-6

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  • DOI: https://doi.org/10.1007/s00041-009-9097-6

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