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2014 Comparing Fréchet and positive stable laws
Thomas Simon
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Electron. J. Probab. 19: 1-25 (2014). DOI: 10.1214/EJP.v19-3058

Abstract

Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha)\alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{ \gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that ${\bf Z}_\alpha^{ \alpha}\,\prec_{st}\, \Gamma(1-\alpha) {\bf L}$ and that $\Gamma(1+\alpha){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_\alpha$ with the two extremal Fréchet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.

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Thomas Simon. "Comparing Fréchet and positive stable laws." Electron. J. Probab. 19 1 - 25, 2014. https://doi.org/10.1214/EJP.v19-3058

Information

Accepted: 28 January 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1288.60018
MathSciNet: MR3164769
Digital Object Identifier: 10.1214/EJP.v19-3058

Subjects:
Primary: 33E12
Secondary: 60E05 , 60E15 , 60G52 , 62E15

Keywords: Convex order , Fréchet distribution , median , Mittag-Leffler distribution , Mittag-Leffler function , stable distribution , stochastic order

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