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

Nuha Altaymani and Wissem Jedidi ()
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Nuha Altaymani: Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Wissem Jedidi: Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2023, vol. 11, issue 19, 1-26

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 E α , 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 ↦ E α z x , 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; infinite divisibility; Laplace transform; Mittag-Leffler function; stable distributions; Stieltjes transforms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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