New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions

Abstract

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.

Keywords

• complete and Thorin Bernstein functions
• complete monotonicity
• generalized Cauchy distribution
• generalized gamma convolutions
• hitting time of spectrally positive stable process
• hyperbolic complete monotonicity