Total progeny for spectrally negative branching L{\'e}vy processes with absorption

Christophe Profeta (LaMME)

REVIEW 2 major objections 3 minor 12 references

Total progeny for spectrally negative branching L{\'e}vy processes with absorption

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For spectrally negative branching Lévy processes killed at zero, the paper proves that the tail of the total number of particles absorbed is of order $n^{-\Phi_+(-\beta)/\Phi_-(-\beta)}$ subcritically and $1/(n\log^2 n)$ at criticality.

desk verdict Subcritical tail asymptotics for spectrally negative branching Lévy processes are solid, but the critical-case upper bound rests on an erroneous Lemma 5 and needs fixing before acceptance. read the letter →

arxiv 2506.04783 v1 pith:YWUCE6ME submitted 2025-06-05 math.PR

Christophe Profeta (LaMME) This is my paper

classification math.PR MSC 60J8060G51

keywords branchingLévyprocessesspectrallynegativetotalprogenytailasymptoticsscalefunctionssubcriticalandcriticalregimesabsorptionatzeroexponentialtilt

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

When a spectrally negative branching Lévy process is killed the moment any particle crosses below zero, and the branching rate is weak enough that the population dies out almost surely, the total number of particles that get absorbed is a heavy-tailed random variable. This paper establishes its tail asymptotics in both regimes. If $\beta

What carries the argument

The load-bearing object is the recursive moment identity obtained by cutting the process at the first branching event and at an upper barrier $x$. The paper defines $Z_{0<x}$ as the number of particles absorbed without any ancestor having reached level $x$, and Lemma 3 gives an exact equation for $E_a[Z_{0<x}^n]$ in terms of the exit event $\{\tau^-_0<\tau^+_x\}$ and an integral over the resolvent of the killed Lévy process; the error term $R_n$ is controlled by $2^{n+1}E_z[Z_{0<x}]E_z[Z_{0<x}^{n-1}]$. The exponential tilt (1.2) moves the Laplace exponent so that the roots $\Phi_-(-\beta),\Phi_+(-\beta)$ (or $\Phi(-q^*)$ at criticality) dictate the size of moments, and the scale functions $W^{(q)}$ defined by (1.1) supply exact prefactors. In the critical case, concavity of $W_{\Phi(-q^*)}$ is used to apply a Tauberian theorem for regularly varying functions and to keep the signs in Lemma 5.

What would settle it

Take a spectrally negative Lévy process with $\Psi'(0+)<0$ and $\beta=q^*$ for which the transformed Lévy measure $\nu_{\Phi(-q^*)}$ has a tail $x\mapsto\nu_{\Phi(-q^*)}(-\infty,-x)$ that is not log-convex, so that the concavity assumption is violated, and compute or simulate $P_a(Z_0\ge n)$ for large $n$ via the recursive moment equations. If the tail deviates from $1/(n\log^2 n)$, the concavity assumption is load-bearing; if the same order appears, the assumption is removable.

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Extended reading notes

Core claim

The central claim is that the boundary-absorption count $Z_0$ has the same polynomial decay as the all-time maximum $M$ of the branching process, with the exponent shifted by the ratio $\Phi_+(-\beta)/\Phi_-(-\beta)$ in the subcritical case and by an extra $\log^2 n$ factor in the critical case. The paper proves matching two-sided bounds: for every $a>0$ there are constants $\kappa_1,\kappa_2>0$, independent of $a$, such that as $n\to\infty$, $\kappa_1 W^{(-\beta)}(a)n^{-\Phi_+(-\beta)/\Phi_-(-\beta)}\le P_a(Z_0\ge n)\le \kappa_2 W^{(-\beta)}(a)n^{-\Phi_+(-\beta)/\Phi_-(-\beta)}$ when $\beta<q^*$, and $\kappa_1 W^{(-q^*)}(a)/(n\log^2 n)\le P_a(Z_0\ge n)\le \kappa_2 W^{(-q^*)}(a)/(n\log^2 n)$ when $\beta=q^*$, the latter under the concavity of $x\mapsto e^{-\Phi(-q^*)x}W^{(-q^*)}(x)$. The $1/(n\log^2 n)$ factor is the trace of the $1/x$ factor in the critical maximum tail.

Load-bearing premise

Everything in the critical case rests on the concavity of $x\mapsto e^{-\Phi(-q^*)x}W^{(-q^*)}(x)$ on $[0,\infty)$; if that concavity fails, the upper bound of Theorem 1(2) is not established, even though the paper describes the assumption as essentially technical.

Editorial extensions

If this is right

In the subcritical regime the tail exponent is the ratio $\Phi_+(-\beta)/\Phi_-(-\beta)$, so the full Laplace exponent of the driving process, not just its drift, controls how slowly $Z_0$ grows.
At criticality the tail $1/(n\log^2 n)$ is heavier than any subcritical power law, matching the intuition that rare huge families are produced by one particle reaching a very high level.
The initial position $a$ enters only through the scale function $W^{(-\beta)}(a)$ or $W^{(-q^*)}(a)$, uniformly in $n$.
The same order holds for the total number of individuals that ever lived, $2Z_0-1$.
For $\beta<q^*$ the proof yields the tail without any concavity assumption; only the critical case is conditional.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the concavity condition is indeed only technical, the critical tail $1/(n\log^2 n)$ should hold for every spectrally negative branching Lévy process with $\beta=q^*$; the natural test is a family of Lévy measures for which the exponentially tilted tail is not log-convex.
The moment-recursion method could be pressed further to yield explicit constants or a distributional limit for $Z_0$ after rescaling, but the paper only establishes the order of the tail.
The same 'tail of progeny from tail of maximum' mechanism suggests a general heuristic for killed branching Markov processes: the exponent of $Z_0$ is fixed by the Laplace exponent, and logarithmic factors appear exactly when the maximum tail has a polynomial correction.
In cases where scale functions and $\Phi_\pm$ are explicit, such as stable-like spectrally negative processes, the theorem gives directly testable predictions for Monte Carlo simulation of $P_a(Z_0\ge n)$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the subcritical part of Theorem 1 is a real result and the proof is careful; the critical part is not proven because Lemma 5(2) has an algebraic error. The reader's conditional verdict is in the right direction, but the condition is worse than the report suggests.

What's new: the paper extends Maillard's branching Brownian motion result and Aïdékon–Hu–Zindy's branching random walk result to spectrally negative branching Lévy processes, and the subcritical exponent Φ+(−β)/Φ−(−β) is natural. The proof strategy is sound: a recursive moment bound, Esscher transform, resolvent identity, and Paley–Zygmund for the lower bound. The critical rate 1/(n log² n) is plausible given the 1/x maximum tail in (1.4). The citation to the author's earlier paper for the maximum asymptotics is appropriate, not circular.

The soft spot: Lemma 5(2) is wrong. The Laplace transform of D_y(a) − 2y/A has limit −y²/A − 2Ψ'''(Φ)y/(3A²), not the displayed −2Ψ'''(Φ)y/(3A²). The paper's monotone convergence argument fails because D_y(a) − 2y/A is negative on (0,y); in the pure quadratic case Ψ(λ)=σ²λ²/2, the integral is exactly −y²/σ². Since Lemma 6 uses Lemma 5(2) to bound the undershoot term H, and Lemma 7 uses Lemma 6, the critical upper bound in §4.3 is unsupported. The concavity assumption itself is also under-verified; calling it "essentially technical" undersells it, since it is doing real work in the Tauberian step.

The rest of the paper reads carefully: minor typos aside, the subcritical proof holds together. The lower bound in the critical case uses Lemma 6 only asymptotically, so it may survive a fix to Lemma 5, but the upper bound needs a new argument.

Who this is for: probabilists working on branching Lévy processes or killed branching processes. The subcritical result is citable now; the critical part needs a corrected proof. Send it to a referee with a request to fix Lemma 5; I would not accept it in this form. Major revision.

Referee Report

2 major / 3 minor

Summary. The paper studies spectrally negative branching L\'evy processes with particles killed upon crossing below zero, and proves tail asymptotics for Z0, the total number of particles absorbed at zero. Under the extinction condition \Psi'(0+)<0 and \beta\le q^*, Theorem 1 gives, as n\to\infty, two-sided bounds of the form \kappa_1 W_{-\beta}(a)n^{-\Phi_+(-\beta)/\Phi_-(-\beta)} in the subcritical case \beta<q^*, and \kappa_1 W_{-q^*}(a)/(n\log^2 n) in the critical case \beta=q^*, assuming concavity of x\mapsto e^{-\Phi(-q^*)x}W_{-q^*}(x). The proof follows A\"id\'ekon's method: recursive moment bounds, Esscher transforms, resolvent identities, and comparison with the previously obtained asymptotics of the branching maximum M from the author's earlier paper [11]. The subcritical case is proved via moment bounds up to an appropriate order and a Paley-Zygmund lower bound; the critical case uses finer scale-function estimates and the concavity assumption.

Significance. If correct, the result is a natural and valuable extension of killed branching random walk and branching Brownian motion results to spectrally negative branching L\'evy processes, with sharp power-law and n^{-1}\log^{-2}n tails in the two regimes. The paper is largely self-contained and gives explicit, checkable arguments in the subcritical case. The critical case is more delicate and rests on a concavity assumption on a scale function; this assumption is stated as a hypothesis, so it is not an internal inconsistency, but it is genuinely load-bearing rather than merely cosmetic. The proof uses classical tools---Tauberian theorems, resolvent kernels, Esscher transforms---and the main ideas are clearly presented. However, an algebraic error in Lemma 5(2) affects the critical-case upper bound and requires correction before the result can be accepted as proved.

major comments (2)

[Section 4.1, Lemma 5(2)] The Tauberian constant in Lemma 5(2) is computed incorrectly. With A=\Psi''(\Phi(-q^*)) and B=\Psi'''(\Phi(-q^*)), since \Psi(\Phi(-q^*)+\lambda)+q^* = A\lambda^2/2 + B\lambda^3/6 + O(\lambda^4), one has \int_0^\infty e^{-\lambda a}(D_y(a)-2y/A)\,da = (1-e^{-\lambda y})/(\Psi(\Phi+\lambda)+q^*) - 2y/(A\lambda) \to -y^2/A - 2By/(3A^2), not -2By/(3A^2). The omitted -y^2/A term comes from the interval a<y, where D_y(a)-2y/A = W_\Phi(a)-2y/A is typically negative; hence the monotone-convergence and positivity argument in the lemma is not valid. In the quadratic case \Psi(\lambda)=\sigma^2\lambda^2/2-\mu\lambda, one has W_\Phi(x)=2x/\sigma^2 and the integral in question is exactly -y^2/\sigma^2, whereas the paper's formula gives 0 and asserts positivity. Because Lemma 5(2) is used to control the undershoot term in Lemma 6, which in turn feeds Lemma 7 and the upper-bound argument in Section 4.3, the critical-case upper bound is not established as written. A corrected expansion together with an additional estimate on \int_y^\infty (D_y(a)-2y/A)\,da, or on the convergence of W_\Phi' to 2/A, is needed.
[Section 4.2, Lemma 6] The application of Lemma 5(2) in the proof of Lemma 6 treats the constants c1,c2 as independent of the shift parameter u and of y, but the statement of Lemma 5(2) does not assert such uniformity. Since Lemma 6 concludes with constants independent of y and integrates the bound over u against the L\'evy measure, the uniformity should either be stated and proved in Lemma 5 or derived explicitly in Lemma 6. This affects the bound (4.3) and therefore the critical-case upper bound.

minor comments (3)

[References] Several names are corrupted by encoding errors: "A\"\i d\'ekon" should be A\"id\'ekon, "Mi/suppress lo\'s" should be Mi\l{}o\'s, and in reference [6] "Goldie and. J L. Teugels" should be "Goldie and Teugels". These should be corrected before publication.
[Remark 2] The concavity assumption is called "essentially technical" in Remark 2, but the proof uses it in an essential way for the non-negativity and monotonicity properties in Lemma 5 and for the sign/control estimates in Lemma 6. The authors should state more explicitly which parts of the critical-case proof would fail without this assumption and perhaps add a brief discussion of known sufficient conditions beyond the cited log-convexity criterion.
[Section 4.1] The sentence "An application of the monotone convergence theorem then shows that its integral over (0,+\infty) is finite" is not accurate as written, because the integrand D_y(a)-2y/A is not non-negative on (0,y). A dominated convergence or direct regular-variation argument is required; this is closely tied to the error in Lemma 5(2) and should be reworked together with it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof combines an independent recursive equation with maximum asymptotics from the author's prior paper used as an external input.

full rationale

After walking the derivation chain, I find no circular step. Theorem 1 is proved by first establishing the recursive moment equation E_a[Z^n_{0<x}] in Lemma 3 from the Markov property, then deriving moment bounds (Lemmas 4, 6, 7) and applying Paley-Zygmund together with the maximum asymptotics (1.3)-(1.4). The maximum asymptotics are imported from the author's prior paper [11], but they concern the different observable M and are stated with their own assumptions; they are used as an external input, not as a renamed or disguised form of the target tail P_a(Z0 ≥ n). The critical-case concavity condition is taken from Kyprianou-Rivero-Song [9] as a stated sufficient condition on W_{Φ(-q*)}, and it is not derived from Theorem 1(2), so it creates no equivalence between input and output. The reviewer's concern about Lemma 5(2) in Section 4.1 is a proof-correctness issue (the displayed Tauberian constant may be missing a term), not a circularity issue: if valid, it would break the critical upper-bound estimate, but it would not make that estimate tautological. Accordingly, the central derivation is self-contained modulo its stated assumptions and external theorems, and no circularity score is warranted.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof relies on standard Lévy process theory and the prior extinction criterion. The only ad-hoc additional assumption is the concavity of the transformed scale function in the critical case, which is explicitly stated as a hypothesis. No new free parameters are fitted to data. No new entities are introduced.

assumptions (4)

domain assumption The spectrally negative Lévy process L has absolutely continuous one-dimensional distributions and is not the negative of a subordinator (Section 1, 'We assume that the one-dimensional distributions...').
This ensures strict convexity of Ψ and guarantees the existence of two roots Φ±(q) for q in the appropriate range. Standard in Lévy process fluctuation theory.
domain assumption The branching process becomes extinct almost surely if and only if Ψ'(0+) < 0 and β ≤ q* (from [11], used throughout).
This sets the regimes studied. The paper cites its own prior work [11] for this condition, which is treated as a known result.
ad hoc to paper In the critical case, the scale function W_{Φ(-q*)}(x) is concave on x ≥ 0 (Theorem 1(2) and Remark 2).
This is a sufficient condition for the Tauberian argument and for sign controls. The paper calls it 'essentially technical' but does not prove it for all cases; it gives a sufficient condition from Kyprianou-Rivero-Song.
standard math The Markov property under Esscher transforms and the resolvent formula (3.4) for the killed process are used, which are standard results in Lévy process theory (see Kyprianou [7]).
These are classical tools for spectrally negative Lévy processes, invoked with references.

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Cite this review

Pith. "Pith review of Total progeny for spectrally negative branching L{\'e}vy processes with absorption." pith.science (2026). https://pith.science/paper/YWUCE6ME

@misc{pith2026250604783,
  author       = {Pith},
  title        = {Pith review of: Total progeny for spectrally negative branching L\'evy processes with absorption},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YWUCE6ME}},
  note         = {Machine review of arXiv:2506.04783}
}

abstract

We consider a spectrally negative branching L{\'e}vy process in which particles are killed upon crossing below zero. It is known that such a process becomes extinct almost surely if the drift toward -$\infty$ is sufficiently strong to counterbalance the reproduction rate. In this note, we study the tail asymptotics of the number of particles absorbed at the boundary during the lifetime of the process, in both the subcritical and critical regimes.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[11]

C. Profeta. Survival and maximum of spectrally negativ e branching L´ evy processes with absorption. Electron. Commun. Probab. 29 (2024), Paper No. 63, 15 pp

work page 2024
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[2]

A¨ ıd´ ekon, Y

E. A¨ ıd´ ekon, Y. Hu and O. Zindy. The precise tail behavior of the total progeny of a killed branching random walk. Ann. Probab. 41 (2013), no. 6, 3786–3878

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[3]

J. Bertoin. Exponential decay and ergodicity of complet ely asymmetric L´ evy processes in a ﬁnite interval.Ann. Appl. Probab., 7, 156–169, 1997

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J. Bertoin. L´ evy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Pres s, Cambridge, 1996

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[5]

Berestycki, E

J. Berestycki, E. Brunet, S. C. Harris and P. Mi/suppress lo´ s. Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift. J. Funct. Anal. , 273, 2107–2143, 2017

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N. H. Bingham, C. M. Goldie and. J L. Teugels. Regular vari ation. Encyclopedia of Mathematics and its Appli- cations, 27, Cambridge University Press, Cambridge, 1989

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[8]

Kyprianou and Z

A.E. Kyprianou and Z. Palmowski. Quasi-stationary dist ributions for L´ evy processes.Bernoulli, 12, 571–581, 2006

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A.E. Kyprianou, V. Rivero and R. Song. Convexity and smoo thness of scale functions and de Finetti’s control problem. J. Theoret. Probab. 23 (2010), no. 2, 547–564

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Roynette, P

B. Roynette, P. Vallois and A. Volpi. Asymptotic behavi or of the hitting time, overshoot and undershoot for some L´ evy processes.ESAIM Probab. Stat. 12 (2008), 58–93. Universit´e P aris-Saclay, CNRS, Univ Evry, Laboratoire de Math ´ematiques et Mod ´elisation d’Evry, 91037, ...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [11]

C. Profeta. Survival and maximum of spectrally negativ e branching L´ evy processes with absorption. Electron. Commun. Probab. 29 (2024), Paper No. 63, 15 pp

work page 2024

[2] [1]

A¨ ıd´ ekon

E. A¨ ıd´ ekon. Tail asymptotics for the total progeny of the critical killed branching random walk. Electron. Com- mun. Probab. 15 (2010), 522–533

work page 2010

[3] [2]

A¨ ıd´ ekon, Y

E. A¨ ıd´ ekon, Y. Hu and O. Zindy. The precise tail behavior of the total progeny of a killed branching random walk. Ann. Probab. 41 (2013), no. 6, 3786–3878

work page 2013

[4] [3]

J. Bertoin. Exponential decay and ergodicity of complet ely asymmetric L´ evy processes in a ﬁnite interval.Ann. Appl. Probab., 7, 156–169, 1997

work page 1997

[5] [4]

J. Bertoin. L´ evy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Pres s, Cambridge, 1996

work page 1996

[6] [5]

Berestycki, E

J. Berestycki, E. Brunet, S. C. Harris and P. Mi/suppress lo´ s. Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift. J. Funct. Anal. , 273, 2107–2143, 2017

work page 2017

[7] [6]

N. H. Bingham, C. M. Goldie and. J L. Teugels. Regular vari ation. Encyclopedia of Mathematics and its Appli- cations, 27, Cambridge University Press, Cambridge, 1989

work page 1989

[8] [7]

Kyprianou

A.E. Kyprianou. Fluctuations of L´ evy processes with applications. Introductory lectures. Second edition. Univer- sitext. Springer, Heidelberg, 2014

work page 2014