Comparing Fréchet and positive stable laws

Abstract

Let $\mathbf{L}$ be the unit exponential random variable and ${\mathbf{Z}}_{\alpha }$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha ){\alpha }^{{\gamma }_{\alpha }}{\mathbf{Z}}_{\alpha }^{-{\gamma }_{\alpha }},0<\alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha ){\mathbf{Z}}_{\alpha }^{-{\gamma }_{\alpha }},0<\alpha <1\}$ is increasing for the convex order, with ${\gamma }_{\alpha }=\alpha /(1-\alpha ).$ We also show that $\{\Gamma (1+\alpha ){\mathbf{Z}}_{\alpha }^{-\alpha },1/2\le \alpha \le 1\}$ is decreasing for the convex order, that ${\mathbf{Z}}_{\alpha }^{-\alpha }{\prec }_{st}\Gamma (1-\alpha )Ł$ and that $Γ(1+\a){\bf Z}_α^{-α} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\mathbf{Z}}_{\alpha }$ 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 ${\mathbf{Z}}_{\alpha }$ and ${\mathbf{Z}}_{\alpha }^{-\alpha }$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\mathbf{Z}}_{\alpha }$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.

To appear in Electronic Journal of Probability

References in corpus (2)

• On some properties of the Mittag-Leffler function $E_{\alpha }(-t^{\alpha })$, completely monotone for $t>0$ with $0<\alpha <1$
• Probability distributions with binomial moments

Cited by in corpus (3)

• On some properties of the Mittag-Leffler function $E_{\alpha }(-t^{\alpha })$, completely monotone for $t>0$ with $0<\alpha <1$
• Local and blowing-up solutions for an integro-differential diffusion equation and system
• Sojourn functionals for spatiotemporal random fields with long-memory