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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Nuha Altaymani & Wissem Jedidi, 2023. "New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions," Mathematics, MDPI, vol. 11(19), pages 1-26, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:19:p:4141-:d:1251857
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    References listed on IDEAS

    as
    1. Lennart Bondesson, 2015. "A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1063-1081, September.
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