Small value probabilities for supercritical branching processes with immigration

TL;DR

This paper investigates the probabilities that the normalized population size of a supercritical Galton-Watson process with immigration takes very small values, revealing how offspring and immigration distributions influence these probabilities.

Contribution

It provides new precise estimates for small value probabilities of the limiting population size in supercritical branching processes with immigration.

Findings

Derived explicit small value probability estimates for the limit ${\\mathcal{W}}$

Characterized the impact of offspring and immigration distributions on these probabilities

Enhanced understanding of the tail behavior of the limiting distribution

Abstract

We consider a supercritical Galton-Watson branching process with immigration. It is well known that under suitable conditions on the offspring and immigration distributions, there is a finite, strictly positive limit ${\mathcal{W}}$ for the normalized population size. Small value probabilities for ${\mathcal{W}}$ are obtained. Precise effects of the balance between offspring and immigration distributions are characterized.