Sampling schemes of multitype continuous-time Bienaym\'e-Galton-Watson trees and limiting critical genealogies

Juan Carlos Pardo; Osvaldo Angtuncio Hern\'andez; Simon C. Harris

arxiv: 2502.04588 · v4 · pith:B7CJVBURnew · submitted 2025-02-07 · 🧮 math.PR

Sampling schemes of multitype continuous-time Bienaym\'e-Galton-Watson trees and limiting critical genealogies

Osvaldo Angtuncio Hern\'andez , Simon C. Harris , Juan Carlos Pardo This is my paper

Pith reviewed 2026-05-23 03:49 UTC · model grok-4.3

classification 🧮 math.PR

keywords multitype branching processesBienaymé-Galton-Watson treesgenealogiessampling schemeslimiting critical genealogiescontinuous-time processesmost recent common ancestors

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The pith

Critical multitype Bienaymé-Galton-Watson processes produce a universal limiting sample genealogy independent of the sampling scheme used.

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

The paper studies genealogies of k sampled particles from a continuous-time multitype Bienaymé-Galton-Watson process under two type-dependent schemes: uniform sampling without replacement within types, and sampling by type-dependent weights. For any fixed sampling time it derives the distribution of most recent common ancestor times, ancestral offspring numbers, and type-dependent ancestral structure. Under the assumption that the process is critical with finite second moments, it shows that as the sampling time tends to infinity conditional on survival of the population, the sample genealogy converges to a single limiting object that does not depend on which of the two schemes is used.

Core claim

Under the assumption that the MBGW process is critical with finite second moments, conditional on survival of the population, a large time limiting sample genealogy emerges which is robust to the sampling scheme used. We identify this universal genealogy to have the same tree structure as the single-type case, and we describe its ancestral type behaviour over scaled-times - this essentially being decoupled from the tree structure except at the times of ancestral splitting events.

What carries the argument

The universal limiting genealogy obtained as sampling time tends to infinity conditional on survival, which carries the same tree structure as the single-type limiting genealogy while its ancestral type process evolves separately except at splits.

If this is right

The limiting tree structure is identical to the single-type limiting genealogy identified in the 2020 single-type case.
Ancestral type labels evolve independently of the branching structure except precisely at the instants when lineages split.
The same limiting genealogy arises whether particles are sampled uniformly within each type or according to fixed type-dependent weights.
The result applies to any fixed number k of sampled particles.

Where Pith is reading between the lines

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

The decoupling of tree shape from type evolution suggests that type frequencies along the genealogy can be studied separately from the topology.
The same limiting object may serve as an approximation for genealogies in near-critical multitype models used in population genetics.
Explicit simulation of the limiting object could be used to test statistical procedures that infer ancestral types from modern samples.

Load-bearing premise

The multitype Bienaymé-Galton-Watson process must be critical and possess finite second moments.

What would settle it

A simulation or explicit calculation showing that the two different sampling schemes produce measurably different limiting genealogies in a critical multitype process with finite second moments would falsify the robustness claim.

read the original abstract

We study the genealogies of samples of $k$ distinguished particles drawn from the population alive at some fixed time in a continuous-time multitype Bienaym\'e-Galton-Watson (MBGW) process under two different type dependent sampling schemes: uniform sampling without replacement within types given a fixed type configuration, and sampling according to type-dependent weights. These schemes complement the uniform sampling at fixed time $T$ considered in Angtuncio, Pardo, C. Harris (2026a) which did not distinguish between sampled types. Under each scheme for a fixed sampling time $T$, we characterise the associated times of most recent common ancestors, ancestral offspring distributions, and type-dependent ancestral structure of the sample genealogy. In addition, under the assumption that the MBGW process is critical with finite second moments, we show that, conditional on survival of the population, a large time limiting sample genealogy emerges which is robust to the sampling scheme used. We identify this universal genealogy to have the same tree structure as the single-type case in C. Harris, Johnston, Roberts (2020), and we describe its ancestral type behaviour over scaled-times - this essentially being decoupled from the tree structure except at the times of ancestral splitting events.

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

The paper adds two type-dependent sampling schemes for multitype BGWT and shows the critical limiting genealogy is robust to them and matches the single-type tree structure.

read the letter

The main thing to know is that under the usual criticality and finite-second-moment assumptions, the large-time limiting sample genealogy from a multitype Bienaymé-Galton-Watson process is the same for the two new type-aware sampling schemes as for uniform sampling, and the tree part is identical to the single-type limit already in Harris, Johnston, Roberts 2020, with type information mostly decoupled except at splits. The paper defines uniform sampling without replacement within types (given fixed type counts) and type-weighted sampling, then works out the finite-time MRCA times, ancestral offspring distributions, and type structure for each scheme. The robustness of the limit across schemes is the clearest new claim. It does this cleanly enough on the surface and extends the earlier uniform-at-fixed-time work without obvious circularity. The soft spots are proportionate: the core tree structure is carried over from the single-type case, so the novelty sits mainly in the type handling and the sampling-scheme independence. The result stays inside the branching-process subfield and rests on standard assumptions that are stated explicitly. Without the proofs in front of me I cannot check derivation details, but the stated conditions and the external reference line up without internal contradiction. This is for specialists already working on multitype branching processes and genealogies. A reader who knows the single-type limits and wants explicit sampling characterizations plus the type-decoupling note will get something concrete from it. It deserves a serious referee because the claims are precise, the extension is well-scoped, and the robustness statement is the sort of thing that can be checked directly in the area.

Referee Report

0 major / 3 minor

Summary. The paper characterizes the genealogies of k distinguished particles sampled at fixed time T from a continuous-time multitype Bienaymé-Galton-Watson process under two type-dependent schemes (uniform sampling without replacement within types given a fixed type configuration, and sampling by type-dependent weights). For each scheme it gives explicit descriptions of MRCA times, ancestral offspring distributions, and type-dependent ancestral structure. Under the additional assumption that the process is critical with finite second moments, it proves that, conditional on survival, the large-T limiting sample genealogy is robust to the choice of sampling scheme, coincides in tree structure with the single-type limit of Harris, Johnston and Roberts (2020), and has ancestral type behaviour that is essentially decoupled from the tree geometry except at splitting times.

Significance. If the stated characterizations and the limiting theorem hold, the work supplies a natural multitype extension of the single-type critical genealogy results, together with a robustness statement across sampling mechanisms and an explicit decoupling of type and tree dynamics. These features strengthen the applicability of the limiting object to type-structured populations and provide concrete descriptions that can be used for further analysis or simulation.

minor comments (3)

[Abstract] The citation 'Angtuncio, Pardo, C. Harris (2026a)' appears in the abstract and introduction; if this is a preprint or in-preparation work, a stable identifier or arXiv number would help readers locate it.
[Introduction] Notation for the two sampling schemes is introduced in the abstract but the precise definitions (especially the type-configuration conditioning for the first scheme) would benefit from an early displayed equation or a short dedicated subsection.
[Abstract] The statement that the limiting type behaviour is 'decoupled from the tree structure except at the times of ancestral splitting events' is clear in the abstract; a brief remark on whether this decoupling survives under the weighted sampling scheme would be useful even if the proof is identical.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on multitype sampling schemes and the limiting critical genealogies, as well as the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

1 steps flagged

Minor self-citation to single-type limit; central robustness claim independent

specific steps

self citation load bearing [Abstract, paragraph 2]
"We identify this universal genealogy to have the same tree structure as the single-type case in C. Harris, Johnston, Roberts (2020)"

The identification invokes a prior result by an overlapping author set. While the citation is not used to forbid alternatives or force the present conclusion, it is the sole external anchor for the tree-structure claim and therefore qualifies as a minor self-citation step (score contribution 2).

full rationale

The paper's core result (large-time limiting genealogy under criticality + finite moments, robust across sampling schemes, with tree structure matching the single-type case) is presented as a direct extension of the external Harris-Johnston-Roberts (2020) result. The only self-citation is to the authors' own 2026a work on one sampling scheme and to the 2020 single-type paper (which shares an author). Neither citation is load-bearing for the new robustness or type-decoupling statements; those are derived under the stated assumptions without reducing to the cited inputs by construction. No fitted parameters, self-definitional loops, or ansatz smuggling appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central limiting claim rests on the domain assumption of criticality plus finite second moments; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The MBGW process is critical with finite second moments
Explicitly required for the large-time limiting sample genealogy to exist and be robust.

pith-pipeline@v0.9.0 · 5765 in / 1337 out tokens · 58813 ms · 2026-05-23T03:49:41.737264+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Harris, Samuel G

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K. B. Athreya and P. E. Ney, Branching processes, Dover Publications, Inc., Mineola, NY, 2004, Reprint of the 1972 original [Springer, New York; MR0373040]. 2047480

work page 2004

[2] [2]

Theory Related Fields 117 (2000), no

Jean Bertoin and Jean-Fran cois Le Gall, The B olthausen- S znitman coalescent and the genealogy of continuous-state branching processes , Probab. Theory Related Fields 117 (2000), no. 2, 249--266. 1771663

work page 2000

[3] [3]

Simon Harris, Samuel G. G. Johnston, and Juan Carlos Pardo, Universality classes for the coalescent structure of heavy-tailed G alton- W atson trees , Ann. Probab. 52 (2024), no. 2, 387--433. 4718398

work page 2024

[4] [4]

Harris, Samuel G

Simon C. Harris, Samuel G. G. Johnston, and Matthew I. Roberts, The coalescent structure of continuous-time G alton- W atson trees , Ann. Appl. Probab. 30 (2020), no. 3, 1368--1414. 4133376

work page 2020

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Theory Related Fields 134 (2006), no

Svante Janson, Limit theorems for triangular urn schemes, Probab. Theory Related Fields 134 (2006), no. 3, 417--452. 2226887

work page 2006

[6] [6]

Samuel G. G. Johnston, The genealogy of G alton- W atson trees , Electron. J. Probab. 24 (2019), Paper No. 94, 35. 4003147

work page 2019

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J. F. C. Kingman, The coalescent, Stochastic Process. Appl. 13 (1982), no. 3, 235--248. 671034

work page 1982

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thesis, Universit\" a t Potsdam; Universit\'e Paris Sud XI, 2010

Sophie P \'e nisson, Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis, Ph.D. thesis, Universit\" a t Potsdam; Universit\'e Paris Sud XI, 2010

work page 2010

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Seneta, Non-negative matrices, Halsted Press [John Wiley & Sons], New York, 1973, An introduction to theory and applications

E. Seneta, Non-negative matrices, Halsted Press [John Wiley & Sons], New York, 1973, An introduction to theory and applications. 389944

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B. A. Sewastjanow, Verzweigungsprozesse, R. Oldenbourg Verlag, Munich-Vienna, 1975, \" U bersetzt aus dem Russischen von Walter Warmuth. 408019

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