Fox-H Densities and Completely Monotone Generalized Wright Functions

Luisa Beghin (), Lorenzo Cristofaro () and José Luís Silva ()
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Luisa Beghin: Sapienza
Lorenzo Cristofaro: University of Luxembourg
José Luís Silva: University of Madeira, Campus da Penteada

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-35

Abstract: 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.

Keywords: Fox-H functions; Complete monotone functions; Generalized wright functions; Mellin-barnes integrals; Laplace transform; 26A48; 33E12; 60E05; 60E10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01391-9

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