Asymptotic behaviour of the S-stopped branching processes with countable state space

TL;DR

This paper investigates the long-term behavior of S-stopped branching processes with countable types, proving that their extinction probability converges to a cyclic function with period 1 under certain conditions.

Contribution

It provides a rigorous analysis of the asymptotic extinction probabilities for countable-type branching processes stopped upon hitting a set S, extending previous results to more general settings.

Findings

Extinction probability converges to a cyclic function with period 1

Results apply to subcritical, indecomposable, noncyclic processes

Provides new insights into the asymptotic behavior of countable-type branching processes

Abstract

he starting process with countable number of types \mu(t) generates a stopped branching process \xi(t). The starting process stops, by falling into the nonempty set S. It is assumed, that the starting process is subcritical, indecomposable and noncyclic. It is proved, that the extinction probability converges to the cyclic function with period 1.