Addendum to: Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics

TL;DR

This addendum extends previous results on negative binomial bounds in supercritical Galton-Watson processes, proving the fractional linear lower bound's validity over the entire [0,1] interval.

Contribution

It generalizes earlier bounds to hold for all x in [0,1], strengthening the theoretical understanding of extinction probabilities in population models.

Findings

The fractional linear lower bound is valid for all x in [0,1].

The result applies to the entire domain, not just up to the extinction probability.

Strengthens previous bounds in Galton-Watson process theory.

Abstract

In this addendum we extend Theorem 4.6 on the negative binomial distribution in `Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics' (Journal of Mathematical Biology 92:40, 2026; arXiv:2503.21403). We prove that the fractional linear lower bound to the negative binomial generating function derived there is indeed valid for every $x\in[0,1]$, and not only for $x\in[0,P^\infty_{\rm NB}]$, where $P^\infty_{\rm NB}$ is the extinction probability of the associated Galton-Watson process.