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

Reinhard B\"urger

arxiv: 2604.04848 · v1 · submitted 2026-04-06 · 🧮 math.PR

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

Reinhard B\"urger This is my paper

Pith reviewed 2026-05-10 19:32 UTC · model grok-4.3

classification 🧮 math.PR

keywords Galton-Watson processesnegative binomial distributiongenerating functionssurvival probabilitiesbranching processesextinction probabilitypopulation genetics

0 comments

The pith

The fractional linear lower bound to the negative binomial generating function holds for every x in [0,1].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This addendum extends a theorem from a prior paper on survival probabilities in supercritical Galton-Watson processes. It proves that a fractional linear lower bound previously derived for the generating function of the negative binomial offspring distribution applies across the full unit interval rather than only up to the extinction probability. A reader would care because the result removes a domain restriction that limited the use of these bounds in calculations of long-term survival and related quantities. The proof keeps the same assumptions on the offspring distribution and the supercritical regime as the original theorem.

Core claim

The addendum establishes that the fractional linear lower bound to the negative binomial generating function, previously shown to hold only for x in [0, P^∞_NB], is valid for every x in [0,1]. This is proven directly under the setup of Theorem 4.6 from the referenced work, confirming the inequality without additional restrictions on the variable.

What carries the argument

The fractional linear lower bound to the negative binomial probability generating function, which supplies a rational-function underestimate usable over the full domain [0,1].

If this is right

Survival probability bounds from the original paper can now be invoked without splitting the domain at the extinction probability.
The same inequality applies directly to any analysis that evaluates the generating function at points beyond the extinction threshold.
Population genetics models built on these branching processes inherit the extended validity for all relevant parameter values.
Case distinctions at the extinction probability become unnecessary in proofs or computations that rely on this lower bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The full-domain result may allow cleaner comparison of the negative binomial case with other offspring distributions whose generating functions already have unrestricted bounds.
Numerical verification routines for branching process models can now test the inequality over the entire interval [0,1] without special handling near the extinction probability.
Similar domain-extension arguments might be applicable to other rational or fractional-linear bounds appearing in related extinction or survival analyses.

Load-bearing premise

The negative binomial offspring distribution and the supercritical regime must match the exact conditions of the original Theorem 4.6.

What would settle it

A concrete counterexample would be a set of negative binomial parameters satisfying the supercritical assumptions together with a numerical value x in (P^∞_NB, 1] at which the true generating function lies below the proposed fractional linear expression.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This addendum extends the fractional linear lower bound on the negative binomial generating function to the full [0,1] interval by direct verification on the remaining part.

read the letter

The main takeaway is that this short addendum proves the lower bound from the author's 2026 paper holds for every x in [0,1], not just up to the extinction probability P^∞_NB. The author plugs the closed-form negative binomial generating function into the inequality and checks it directly on (P^∞_NB, 1]. That step uses the same supercritical Galton-Watson setup and offspring distribution as Theorem 4.6 in the prior work, with no added assumptions. The verification is mechanical but closes the domain gap cleanly. In population genetics applications, this lets the bound apply uniformly without splitting cases at extinction, which can simplify some survival probability calculations. The algebra carries over without circularity or division issues, and the stress-test confirms no hidden domain violations. The work stays narrow. It introduces no new methods, no broader theory, and no independent predictions. The paper is brief because the extension amounts to checking one inequality over the extra interval once the generating function is written out. No simulations or data appear, just the algebraic check. This note is useful mainly to readers who already know the original paper and want the updated statement for their calculations. Someone outside branching processes or population genetics would get little from it alone. The thinking is clear and the engagement with the prior result is honest. I would send it to peer review. The claim is precise, the verification is direct, and a referee can confirm the steps quickly. It is minor but keeps the record accurate in a technical area where exact domains matter.

Referee Report

0 major / 2 minor

Summary. This addendum extends Theorem 4.6 of the cited prior paper (arXiv:2503.21403) by proving that the fractional linear lower bound on the negative binomial probability generating function holds for all x in [0,1] rather than only on [0, P^∞_NB]. The argument proceeds by direct verification of the inequality on the interval (P^∞_NB, 1] using the explicit closed-form expression of the generating function, while retaining the supercritical regime and negative binomial offspring distribution from the original theorem.

Significance. If the extension holds, it removes an artificial domain restriction from the survival-probability bounds, thereby widening their applicability to population-genetics models without introducing new parameters or assumptions. The manuscript supplies a parameter-free, direct verification that strengthens the original result.

minor comments (2)

The abstract and introduction could explicitly reference the equation number of the fractional-linear bound from the original Theorem 4.6 to help readers locate the precise statement being extended.
A one-sentence remark on why the closed-form verification does not encounter singularities or sign changes on (P^∞_NB, 1] would improve readability for readers unfamiliar with the negative-binomial PGF.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the addendum. The work indeed removes the artificial restriction to [0, P^∞_NB] by direct verification on the complementary interval.

Circularity Check

0 steps flagged

Direct verification of extended bound; no circularity

full rationale

The addendum extends Theorem 4.6 from the cited prior paper by directly verifying that the fractional linear lower bound on the negative binomial PGF holds on the full interval [0,1] rather than only up to P^∞_NB. This verification uses the explicit closed-form expression of the generating function to check the inequality on (P^∞_NB,1]. The argument carries over the offspring distribution and supercritical regime from the prior work but introduces an independent algebraic check; no fitted parameters are renamed as predictions, no self-definitional reduction occurs, and the central claim does not collapse to a self-citation chain. The result is self-contained against the explicit functional form.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The addendum depends entirely on the setup, definitions, and assumptions of the prior paper (negative binomial offspring distribution, supercritical Galton-Watson process, extinction probability P^∞_NB). No new free parameters, axioms, or invented entities are introduced.

pith-pipeline@v0.9.0 · 5399 in / 991 out tokens · 53735 ms · 2026-05-10T19:32:09.722102+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

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

[1] Bürger R. Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics. Journal of Mathematical Biology 92, 40 (2026) 9

work page 2026

[1] [1]

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

[1] Bürger R. Bounds for survival probabilities in supercritical Galton-Watson processes and applications to population genetics. Journal of Mathematical Biology 92, 40 (2026) 9

work page 2026