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Fox-H Densities and Completely Monotone Generalized Wright Functions

Author

Listed:
  • Luisa Beghin

    (Sapienza)

  • Lorenzo Cristofaro

    (University of Luxembourg)

  • José Luís Silva

    (University of Madeira, Campus da Penteada)

Abstract

Due to their flexibility, Fox-H functions are widely studied and applied to many research topics, such as astrophysics, statistical mechanics, and probability. Well-known special cases of Fox-H functions, such as Mittag-Leffler and Wright functions, find a wide application in the theory of stochastic processes, anomalous diffusions and non-Gaussian analysis. In this paper, we focus on certain explicit assumptions that allow us to use the Fox-H functions as densities. We then provide a subfamily of the latter, called Fox-H densities with all moments finite, and give their Laplace transforms as entire generalized Wright functions. The class of random variables with these densities is proven to possess a monoid structure. We present eight subclasses of special cases of such densities (together with their Laplace transforms) that are particularly relevant in applications, thanks to their probabilistic interpretation. To analyze the existence conditions of Fox-H functions as well as their sign, we derive asymptotic results and their analytic extension.

Suggested Citation

  • Luisa Beghin & Lorenzo Cristofaro & José Luís Silva, 2025. "Fox-H Densities and Completely Monotone Generalized Wright Functions," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-35, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01391-9
    DOI: 10.1007/s10959-024-01391-9
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    References listed on IDEAS

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    1. Kadankova, Tetyana & Simon, Thomas & Wang, Min, 2020. "On some new moments of Gamma type," Statistics & Probability Letters, Elsevier, vol. 165(C).
    2. Gupta, Neha & Kumar, Arun, 2022. "Inverse tempered stable subordinators and related processes with Mellin transform," Statistics & Probability Letters, Elsevier, vol. 186(C).
    3. Jean-Francois Chamayou & Gérard Letac, 1999. "Additive Properties of the Dufresne Laws and Their Multivariate Extension," Journal of Theoretical Probability, Springer, vol. 12(4), pages 1045-1066, October.
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