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REGENERATIVE MUTATION PROCESSES RELATED TO THE SELFDECOMPOSABILITY OF SIBUYA DISTRIBUTIONS

Published online by Cambridge University Press:  22 June 2018

Thierry Huillet  and
Servet Martínez
Thierry Huillet
Affiliation:
Laboratoire de Physique Théorique et Modélisation CNRS UMR-8089, Site de Saint Martin, 2 avenue Adolphe-Chauvin, Université de Cergy-Pontoise Cergy-Pontoise 95302, France E-mail: Thierry.Huillet@u-cergy.fr
Servet Martínez
Affiliation:
Depto. Ingenieria Matematica and Centro Modelamiento Matematico Universidad de Chile UMI 2071, Uchile-Cnrs, Santiago, Casilla 170-3 Correo 3, Chile
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Abstract

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The Sibuya distribution is a discrete probability distribution on the positive integers which, while Poisson-compounding it, gives rise to the discrete-stable distribution of Steutel and van Harn. We first address the question of the discrete self-decomposability of Sibuya and Sibuya-related distributions. Discrete self-decomposable distributions arise as limit laws of pure-death branching processes with immigration, translating a balance between immigration events and systematic ageing and ultimate death of the immigrants at constant rate. Exploiting this fact, we design a new Luria–Delbrück-like model as an intertwining of a coexisting two-types (sensitive and mutant) population. In this model, a population of sensitive gently grows linearly with time. Mutants appear randomly at a rate proportional to the sensitive population size, very many at a time and with Sibuya-related distribution; each mutant is then immediately subject to random ageing and death upon appearance. The zero-set of the times free of mutants, when the sensitive population lacks immunity, is investigated using renewal theory. Finally, assuming each immigrant to die according to a critical binary branching processes, now with heavy-tailed extinction times, we observe that the local extinction events can become sparse, leading to a congestion of the mutants in the system.

Keywords

discrete self-decomposabilityheavy-tailsmutation model of viral resistanceregenerative processesSibuya-related distributions
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 2 , April 2019 , pp. 291 - 325
Copyright
Copyright © Cambridge University Press 2018 

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