Complete left-tail asymptotic for branching processes with immigration

TL;DR

This paper derives a comprehensive asymptotic series for the density of the martingale limit in Galton-Watson processes with immigration, providing the first complete characterization of their left-tail behavior and an efficient approximation method.

Contribution

It introduces the first complete left-tail asymptotic series for the density of the martingale limit in branching processes with immigration, valid everywhere.

Findings

Series converges for all arguments, not just small ones.

Provides a quickly computable approximation for the density.

First complete result on left tails of branching processes with immigration.

Abstract

We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a Galton-Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of branching processes with immigration. A good, quickly computed approximation for the density will also be derived from the series.