Electronic Journal of Probability

Comparing Fréchet and positive stable laws

Thomas Simon

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 16, 25 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33E12: Mittag-Leffler functions and generalizations
Secondary: 60E05: Distributions: general theory 60E15: Inequalities; stochastic orderings 60G52: Stable processes 62E15: Exact distribution theory

Convex order Fréchet distribution Median Mittag-Leffler distribution Mittag-Leffler function stable distribution stochastic order

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Simon, Thomas. Comparing Fréchet and positive stable laws. Electron. J. Probab. 19 (2014), paper no. 16, 25 pp. doi:10.1214/EJP.v19-3058. https://projecteuclid.org/euclid.ejp/1465065658

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