Tails in a fixed-point problem for a branching process with state-independent immigration

TL;DR

This paper analyzes the tail behavior of solutions to a fixed-point equation in branching processes with immigration, especially under heavy-tailed distributions, and explores various generalizations of the model.

Contribution

It provides new tail asymptotics for solutions under heavy-tailed, dominantly varying distributions, extending previous results and considering multiple model generalizations.

Findings

Tail asymptotics characterized for heavy-tailed distributions

Solutions exhibit long-tail behavior under certain conditions

Model generalizations broaden applicability of results

Abstract

We consider a fixed-point equation for a non-negative integer-valued random variable, that appears in branching processes with state-independent immigration. A similar equation appears in the analysis of a single-server queue with a homogeneous Poisson input, feedback and permanent customer(s). It is known that the solution to this equation uniquely exists under mild first and logarithmic moments conditions. We find further the tail asymptotics of the distribution of the solution when the immigration size and branch size distributions are heavy-tailed. We assume that the distributions of interest are dominantly varying and have a long tail. This class includes, in particular, (intermediate, extended) regularly varying distributions. We consider also a number of generalisations of the model.