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

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.

Theorems & Definitions (5)

• Theorem 1
• proof : Proof of the inequality \ref{['varphiNB_ge_varphiFL']} if $x\in[0,P_{\rm{NB}}^\infty]$ (this was already proved in Buerger2026)
• proof : Proof of the inequality \ref{['varphiNB_ge_varphiFL']} if $x\in(P_{\rm{NB}}^\infty,1]$
• Lemma 1
• proof : Proof of Lemma \ref{['lem:cgt']}