REGENERATIVE MUTATION PROCESSES RELATED TO THE SELFDECOMPOSABILITY OF SIBUYA DISTRIBUTIONS

• Published in: Probability in the engineering and informational sciences Vol. 33; no. 2; pp. 291 - 325
• Main Authors: Huillet, Thierry, Martínez, Servet
• Format: Journal Article
• Language: English
• Published: New York, USA Cambridge University Press 01.04.2019; Cambridge University Press (CUP)
• Subjects: Branching (mathematics); Death; Decomposition; Extinction; Immigration; Immunity; Integers; Markov processes; Mathematics; Mutation; Population; Probability; Sibuya-related distributions; heavy-tails; mutation model of viral resistance; discrete self-decomposability; regenerative processes
• ISSN: 0269-9648; 1469-8951
• DOI: 10.1017/S0269964818000189

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.