Scaling features of two special Markov chains involving total disasters

TL;DR

This paper analyzes two special Markov chain models with total disasters, exploring their recurrence, transience, and critical behaviors, and establishing connections to well-known distributions with detailed probabilistic properties.

Contribution

It introduces exactly solvable catastrophe Markov chain models involving total disasters, analyzing their recurrence, invariant measures, and distributional properties, linking them to Sibuya and Pareto-Zipf distributions.

Findings

Models exhibit recurrence/transience transitions depending on parameters

Explicit invariant measures and distributional properties are derived

Connections to extended Sibuya and Pareto-Zipf distributions are established

Abstract

Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto-Zipf distributions whose divisibility and self-decomposability properties are shown relevant. The study includes: existence and shape of the invariant measure, time-reversal, return time to the origin, contact probability at the origin, extinction probability, height and length of the excursions, a renewal approach to the fraction of time spent in the catastrophic state, scale function, first time to collapse and first-passage times, divisibility properties.