Reduced branching processes with very heavy tails

TL;DR

This paper investigates the asymptotic behavior of reduced Markov branching processes with very heavy-tailed reproduction distributions, extending understanding beyond the well-studied critical case to new heavy-tail regimes.

Contribution

It introduces a new asymptotic pattern for the process conditioned on non-extinction under very heavy tails with Zubkov's regularity, expanding theoretical knowledge.

Findings

New asymptotic pattern identified for heavy-tailed regimes

Extended understanding of process behavior when tail index is zero

Results applicable to genealogical models in biology

Abstract

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $\alpha\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $\alpha=0$ assuming Zubkov's regularity condition with parameter $\beta\in(0,\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.