More on the asymptotic behaviour of moments of branching Markov processes

Christopher B. C. Dean; Emma Horton

REVIEW 3 major objections 5 minor 12 references

More on the asymptotic behaviour of moments of branching Markov processes

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Moment limits are explicit for branching Markov processes.

desk verdict A serious extension of moment asymptotics for branching Markov processes to Jordan-block spectra, with clean statements but proofs that stop at k=2 in two of the three main theorems. read the letter →

arxiv 2501.19268 v1 pith:ZSSZ3KZU submitted 2025-01-31 math.PR

Christopher B. C. Dean , Emma Horton This is my paper · ORCID

classification math.PR MSC 60J8060F05

keywords branchingMarkovprocessmomentsnon-localfluctuationsgeneralizedeigenfunctiondecompositionlargeandsmallregimescentrallimittheoremspatialprocesses

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

This paper aims to pin down the long-time behaviour of all moments of a branching Markov process with non-local branching. It claims that once the linear expectation semigroup admits a generalized Perron–Frobenius decomposition that allows non-simple leading eigenvalues, every $\ell$-th moment $\psi_t^{(\ell)}[f](x)$, properly rescaled, converges uniformly to a limit $L_{[\ell]}[f](x)$ that is given explicitly in terms of lower-order moments. The normalization and the limit take three different forms depending on whether the relevant eigenvalue is large, small, or critical relative to the leading eigenvalue. If the results are correct, they supply the moment asymptotics needed for central limit theorems and for new proofs of laws of large numbers for spatial branching processes.

What carries the argument

The engine is the generalized eigenelement decomposition (H1) of the linear expectation semigroup $\psi_t$: there are finitely many eigenvalues $\lambda_i$, with $\lambda_1$ real and dominant, an operator $N$ acting nilpotently along each chain, eigenfunctions and generalized eigenfunctions $\phi_{i,j}^{(k)}$, and dual functionals such that $\psi_t$ acts as $e^{(\lambda_i+N)t}$ on each chain, with a uniformly decaying remainder after subtracting the chains. This permits Jordan-chain structure at the leading eigenvalue rather than only the simple Perron–Frobenius form. The second ingredient is the partition operator $\zeta_A$ and the evolution equation of Lemma 4.1, $\psi_t^{(k)}[f](x)=\psi_t[\prod_{i=1}^k f_i](x)+\int_0^t \psi_s[\gamma\zeta_{[k]}[\psi_{t-s}^{(\cdot)}[f]]](x)\,ds$, which expresses a $k$-th moment as lower-order moments and drives the induction. The limit objects $L_{[\ell]}$ are the fixed points of this integral recursion.

What would settle it

A concrete check is to build a branching Markov process whose expectation semigroup has a non-simple leading eigenvalue, verify whether (H1) holds, and compare Monte Carlo estimates of the rescaled second moment with $L_{[2]}$ from Theorem 2.4. A failure of the uniform remainder bound in (1.3), or a mismatch in the time power $(1+t)^{-(p(f)+\ell(p_1-2)/2)}$, would falsify the critical-regime theorem; if no such process satisfying (H1) can be constructed, the theorems concern an empty class.

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Extended reading notes

Core claim

The central claim is Theorems 2.1, 2.2, and 2.4: for $1 \le \ell \le k$, under the spectral assumption (H1) and the moment condition $\sup_x E_x[N^k] < \infty$, and for initial functions lying in the large, small, or critical regime, the normalized moment converges uniformly as $t \to \infty$. In the large regime $e^{-\tilde\lambda(f)t}(1+t)^{\tilde p(f)-\ell}\psi_t^{(\ell)}[f](x)$ converges to $\prod_{j=1}^{\ell}(p(f_j)-1)!\, L_{[\ell]}[f^*](x)$; in the small regime $e^{-\ell\lambda_1/2}(1+t)^{-\ell(p_1-1)/2}\psi_t^{(\ell)}[f](x)$ converges to $L_{[\ell]}[f](x)$; and in the critical regime $e^{-\ell\lambda_1/2}(1+t)^{-(p(f)+\ell(p_1-2)/2)}\psi_t^{(\ell)}[f](x)$ converges to $L_{[\ell]}[f](x)$. The limits are built from the eigenprojection $\Phi$, the operator $N$, and integrals of lower-order moment limits along the semigroup evolution.

Load-bearing premise

Everything rests on assumption (H1): the expectation semigroup must decompose into finitely many Jordan chains of eigenfunctions with an operator $N$ and a uniform remainder that decays like $e^{\mathrm{Re}\,\lambda_m t}$, and the paper does not give a concrete example of a non-simple-eigenvalue branching Markov process that satisfies this decomposition.

Editorial extensions

If this is right

The results extend prior moment asymptotics to branching Markov processes whose leading eigenvalue is not simple, so models with degenerate or Jordan-type spectra are covered.
The explicit limits reduce the verification of moment asymptotics for a given model to checking (H1) and the finite-$k$-th-moment condition, after which all moments up to order $k$ are determined recursively.
The critical-regime normalization shows that the fluctuation rate contains the Jordan-chain data $p(f)$ and $p_1$, so the time power in the scaling changes when the spectral decomposition has generalized eigenvectors.
The same proof machinery yields convergence of odd moments under a different scaling, as noted in Remarks 2.3 and the remark after Theorem 2.4.
The moment limits are the ingredient the authors identify as needed for proving central limit theorems and for deriving new proofs of laws of large numbers for spatial branching processes.

Reading between the lines

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

Editorial extension: the stated second-moment limits point toward a spatial central limit theorem in which the centred empirical measure converges to a Gaussian field with covariance encoded by $L_{[2]}$; the paper does not prove this convergence, but the moment controls it provides are the missing ingredient.
Editorial extension: the paper does not exhibit a concrete branching Markov process with a non-simple leading eigenvalue that satisfies (H1), so a natural test is to verify (H1) for a model such as branching Brownian motion on a compact state space where eigenvalues can be degenerate.
Editorial extension: the induction through $\zeta_A$ should also apply to joint moments of several test functions and to genealogical many-to-few maps, so the method may give joint asymptotic normality rather than only marginal moment limits.

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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

The paper does something genuinely useful: it takes the moment asymptotics of [7] and extends them to settings where the leading eigenvalue is not simple, allowing generalized eigenfunctions. The trichotomy into large, critical, and small regimes is natural, and the limit formulas, including the factorial constants in the critical case, are explicit and plausible. The proofs for k=2 in all three regimes are spelled out, and the base case is solid. This is a meaningful extension, not a repackaging.

The soft spots are real. Theorems 2.2 and 2.4 state results for every ℓ up to k, but the inductive step for ℓ≥3 is dismissed with 'identical structure' and omitted. In the critical regime that is not obviously identical: odd ℓ sits outside the conjugate-tuple definition, and the limit is set to 0 without a genuine argument. The authors' own remark that odd moments require a different scaling suggests the present normalization is not the natural one for odd ℓ, so the assertion L=0 deserves a proof. The stress-test's specific claim that odd-ℓ terms diverge like t^{p1/2} does not survive a back-of-envelope: the exponential factor in the evolution equation suppresses the extra polynomial growth, and a crude estimate gives the odd-ℓ contribution of order t^{-p1/2} relative to the normalization. So the statement may be true, but it is not demonstrated, and the missing induction is a genuine gap.

Two more issues. Lemma 4.1, the evolution equation that drives all proofs, is only sketched; a referee will want to check the sign of the exponential and the partitioning carefully, since the proof in Section 3 uses a form that is a mirror image of the lemma. And (H1) is a strong assumption: no concrete branching Markov process with a non-simple leading eigenvalue is shown to satisfy it. That leaves the applicability of the main results open, even if the spectral framework is natural.

Minor: the theorems say 'for 1≤ℓ≤k' but the normalized quantities are defined only for 2≤ℓ≤k.

Bottom line: the paper is worth refereeing, but the referees should demand the full induction or a reduction of the claims to the k=2 case, and a discussion of models satisfying (H1). The audience is specialists in branching processes and spectral methods; for them this is a credible step toward CLTs. I'd send it out, but with a clear request for the missing proofs.

Referee Report

3 major / 5 minor

Summary. The paper studies the asymptotic behaviour of the moments of a branching Markov process with non-local branching, under a general Perron--Frobenius-type assumption (H1) that allows the leading eigenvalue to be non-simple and that also tracks further eigenvalues. The main results, Theorems 2.1, 2.2, and 2.4, claim that for every order 1 ≤ ℓ ≤ k and every initial test function in the large, small, or critical spectral regime, the suitably normalised moment e^{-scale}ψ_t^{(ℓ)}[f](x) converges uniformly in x and f to an explicit limit L_[ℓ][f](x). The proofs are built on an evolution equation (Lemma 4.1), a base case at k=2 for the small and critical regimes, and an inductive step that is asserted to follow the structure of the large-regime proof but is not written out. The paper also states two remarks indicating that odd moments require a different scaling to obtain non-zero limits.

Significance. If the results are correct, they give a substantial generalisation of the moment asymptotics of Gonzalez--Horton--Kyprianou [7] to non-simple leading eigenvalues, and they supply the higher-order fluctuation estimates needed for central limit theorems and laws of large numbers for branching Markov processes. The paper contains detailed proofs of the base cases k=2 for the small and critical regimes, and the large-regime proof is mostly complete. However, the central theorems as stated contain concrete algebraic errors in the limiting constants, and the inductive steps for all ℓ≥3 in the small and critical regimes are omitted. These issues prevent acceptance in the current form, but they appear fixable within the manuscript's scope.

major comments (3)

[§2, Theorem 2.1 and Theorem 2.2] The limiting constants in the two theorems have inverted factorial factors. In Theorem 2.1 the stated limit is the product ∏_{j=1}^ℓ (p(f_j)-1)! L_{[ℓ]}[f*](x), but the proof in §3.1, together with the ℓ=1 consequence of (H1), yields ∏_{j=1}^ℓ 1/(p(f_j)-1)! L_{[ℓ]}[f*](x). In Theorem 2.2, the |A|=2 definition of L_A carries a factor (p1-1)! in the numerator, whereas the derivation from Lemma 3.2 and the asymptotic e^{Nt} ~ t^{p1-1}/(p1-1)! N^{p1-1} gives the reciprocal. For p(f)≥3 or p1≥3 the stated limits are wrong by a factorial-squared factor, so the central claims are not correct as written.
[§3.2 and §3.3, proofs of Theorems 2.2 and 2.4] The inductive steps for ℓ≥3 are asserted to follow by an 'identical structure' to the large-regime proof and are omitted. This is load-bearing: the theorems claim convergence for every 1≤ℓ≤k, and in the critical regime the partitions of [ℓ] for odd ℓ contain an unpaired critical singleton alongside conjugate pairs; the polynomial exponents must be traced through the integral in Lemma 4.1 to show that the normalisation in (2.4) indeed forces those contributions to vanish. The stress-test concern that these contributions grow faster than the normalisation does not appear to land --- the extra factor e^{-(ℓ/2-1)λ1 s} in the s-integral makes the integral dominated by s=O(1), so the odd-ℓ contributions vanish under the stated scaling --- but the required estimates are not supplied. The authors should provide the full induction or a rigorous outline showing how the large-regime argument adapts to the possible 2+...+2+1 partitions.
[§4, Lemma 4.1] The evolution equation (4.2) is the backbone of all three main theorems, but its proof is only a sketch referring to [9, Proposition 9.1]. Since the present setting includes non-local branching, a more general state space, and complex-valued test functions, the text should either give a complete proof of (4.2) or state precisely which steps in [9] carry over unchanged and which require modification. As written, an independent check of the main results is not possible without reconstructing this lemma.

minor comments (5)

[§2, Theorem 2.4] In (2.4) and in the definition of L_A, the notation p(f) is used where the vector quantity ~p(f)=Σ_i p(f_i) is intended; Lemma 3.3 uses the correct ~p(f) in β(f). The same missing tilde affects the factor (p(f)-2)! in the |A|=2 formula.
[§2, Theorem 2.1, equation (2.1)] In the definition of L_A, the exponential e^{-~λ(f)s} should presumably be e^{-~λ(f,A)s}, since the integral is over subsets A whose growth rate is governed by the sum of λ(f_i) for i∈A; the notation ~λ(f,A) introduced in §3 supports this reading.
[§1.1 and §2] The paper motivates (H1) by saying that many processes have a non-simple leading eigenvalue, but it gives no concrete example or reference to a process satisfying (H1) with p1≥2 or with m≥2. A short example or citation would make the assumptions more tangible and strengthen the paper's applicability.
[Remarks after Theorems 2.2 and 2.4] The remarks stating that odd moments can be handled under a different scaling are useful, but they should explicitly clarify that they do not contradict the L_[ℓ]=0 statements in the current theorems; under the normalisations in (2.3) and (2.4) the odd moments are claimed to vanish, and the different scaling is only needed for non-zero odd-order limits.
[Abstract] There is a minor typographical error in the first sentence ('result s' instead of 'results').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are conditional on an explicit spectral assumption and the limit objects are defined constructively via lower-order moments.

full rationale

The paper derives moment asymptotics under the stated assumption (H1), which is a generalized Perron–Frobenius condition on the expectation semigroup, not on the moments being proved. The limit functions L_A in Theorems 2.1, 2.2, and 2.4 are defined by explicit integrals involving ψ_s and ζ_A[L·], where ζ_A only uses moments of strictly smaller order; this is a well-founded inductive definition, not a restatement of the target scaling. The evolution equation (4.2) is attributed to the authors' prior work [9, Proposition 9.1] and [9, Theorem 8.2], but it is a general identity for branching Markov processes, not the asymptotic claim, and a proof sketch is provided. The proof strategy is said to follow [7], but that is a methodological borrowing, not a logical dependence: the results extend [7] to non-simple leading eigenvalues, a new setting not established there. No parameter is fitted to data, no 'prediction' is constructed from the quantity it predicts, and no uniqueness theorem from the authors' prior work is invoked to force a choice. The omission of the inductive step for k≥3 in the proof of Theorem 2.4 is a completeness flaw, but it does not constitute circular reasoning. The paper is self-contained relative to its assumptions; any concern about the validity of (H1) or the correctness of the critical-regime statement is a mathematical risk, not a circularity issue.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; (H1) is an abstract spectral assumption that carries the whole burden. The only imported tools are standard semigroup estimates and a previously known evolution equation.

assumptions (5)

domain assumption Assumption (H1): the expectation semigroup ψ_t admits a finite generalized eigenfunction decomposition with Jordan operator N and uniform error bound (1.3).
Stated in Section 1.1 and assumed in force before all main theorems; no concrete process is shown to satisfy it.
domain assumption Moment condition sup_{x∈E} E_x[N^k] < ∞ for the k under consideration.
Assumed in Theorems 2.1, 2.2, 2.4 and Lemma 4.1; excludes heavy-tailed offspring numbers.
domain assumption Branching rate γ is bounded and offspring distribution is given by a probability kernel P_y.
Setup in Section 1; boundedness is used throughout the estimates.
standard math Evolution equation Lemma 4.1, extending [9, Proposition 9.1], connecting k-th moments to lower-order moments.
Used as the inductive engine; proof is only sketched and relies on [9, Theorem 8.2].
standard math Perron-Frobenius style convergence and standard analytic estimates (e.g., (3.1), (3.2)) for the semigroup generated by λ_i and N.
Invoked throughout Section 3 to bound remainders and exchange integrals.

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Cite this review

Pith. "Pith review of More on the asymptotic behaviour of moments of branching Markov processes." pith.science (2026). https://pith.science/paper/ZSSZ3KZU

@misc{pith2026250119268,
  author       = {Pith},
  title        = {Pith review of: More on the asymptotic behaviour of moments of branching Markov processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSSZ3KZU}},
  note         = {Machine review of arXiv:2501.19268}
}

abstract

Consider a branching Markov process, $X = (X(t), t \ge 0)$, with non-local branching mechanism. Studying the asymptotic behaviour of the moments of X has recently received attention in the literature [6, 7] due to the importance of these results in understanding the underlying genealogical structure of $X$. In this article, we generalise the results of [7] to allow for a non-simple leading eigenvalue and to also study the higher order fluctuations of the moments of $X$. These results will be useful for proving central limit theorems and extending well-known LLN results.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[7]

Gonzalez, E

I. Gonzalez, E. Horton, and A. E. Kyprianou. Asymptotic momen ts of spatial branching processes. Probability Theory and Related Fields, 184(3-4):805–858, 2022

work page 2022
[9]

Horton and A

E. Horton and A. E. Kyprianou. Stochastic neutron transport and non-local branching Markov processes. Probability and its Applications. Birkh¨ auser, 2023

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[1]

Asmussen and H

S. Asmussen and H. Hering. Strong limit theorems for general su percritical branching processes with applications to branching diﬀusions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36(3):195–212, 1976

work page 1976
[2]

Limit theorems for the neutron transport equation

Eric Dumonteil, Emma Horton, Andreas E Kyprianou, and Andrea Z ola. Limit theorems for the neutron transport equation. arXiv preprint arXiv:2407.04820, 2024

work page Pith review arXiv 2024
[3]

S.D. Durham. Limit theorems for a general critical branching pro cess. Journal of Applied Probability, 8(1):1–16, 1971

work page 1971
[4]

Engl¨ ander, S.C

J. Engl¨ ander, S.C. Harris, and A. E. Kyprianou. Strong law of lar ge numbers for branching diﬀusions. Ann. Inst. Henri Poincar´ eProbab. Stat., 46(1):279–298, 2010

work page 2010
[5]

Fleischman

J. Fleischman. Limiting distributions for branching random ﬁelds. Trans. Amer. Math. Soc., 239:353–389, 1978

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[6]

Convergence of genealogies through spinal decomposition with an application to population genetics

F´ elix Foutel-Rodier and Emmanuel Schertzer. Convergence of genealogies through spinal decomposition with an application to population genetics. Probability Theory and Related Fields, 187(3):697–751, 2023

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[8]

Many-to-few for non-local branching markov process

Simon C Harris, Emma Horton, Andreas E Kyprianou, and Ellen Powe ll. Many-to-few for non-local branching markov process. Electronic Journal of Probability, 29:1–26, 2024

2024
[10]

Stability of (sub) critical non-local spatial branching processes w ith and without immigration

Emma Horton, Andreas E Kyprianou, Pedro Mart ´ ın-Ch´ avez,Ellen Powell, and Victor Rivero. Stability of (sub) critical non-local spatial branching processes w ith and without immigration. arXiv preprint arXiv:2407.05472, 2024

2024 arXiv
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I. Iscoe. On the supports of measure-valued critical branch ing Brownian motion. Ann. Probab., 16(1):200–221, 1988

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A. Klenke. Multiple scale analysis of clusters in spatial branching m odels. Ann. Probab., 25(4):1670–1711, 1997. 13

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [7]

Gonzalez, E

I. Gonzalez, E. Horton, and A. E. Kyprianou. Asymptotic momen ts of spatial branching processes. Probability Theory and Related Fields, 184(3-4):805–858, 2022

work page 2022

[2] [9]

Horton and A

E. Horton and A. E. Kyprianou. Stochastic neutron transport and non-local branching Markov processes. Probability and its Applications. Birkh¨ auser, 2023

work page 2023

[3] [1]

Asmussen and H

S. Asmussen and H. Hering. Strong limit theorems for general su percritical branching processes with applications to branching diﬀusions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36(3):195–212, 1976

work page 1976

[4] [2]

Limit theorems for the neutron transport equation

Eric Dumonteil, Emma Horton, Andreas E Kyprianou, and Andrea Z ola. Limit theorems for the neutron transport equation. arXiv preprint arXiv:2407.04820, 2024

work page Pith review arXiv 2024

[5] [3]

S.D. Durham. Limit theorems for a general critical branching pro cess. Journal of Applied Probability, 8(1):1–16, 1971

work page 1971

[6] [4]

Engl¨ ander, S.C

J. Engl¨ ander, S.C. Harris, and A. E. Kyprianou. Strong law of lar ge numbers for branching diﬀusions. Ann. Inst. Henri Poincar´ eProbab. Stat., 46(1):279–298, 2010

work page 2010

[7] [5]

Fleischman

J. Fleischman. Limiting distributions for branching random ﬁelds. Trans. Amer. Math. Soc., 239:353–389, 1978

work page 1978

[8] [6]

Convergence of genealogies through spinal decomposition with an application to population genetics

F´ elix Foutel-Rodier and Emmanuel Schertzer. Convergence of genealogies through spinal decomposition with an application to population genetics. Probability Theory and Related Fields, 187(3):697–751, 2023

work page 2023