Large time behavior of critical marked Hawkes processes with heavy tailed marks and related branching particle systems

Anna Talarczyk

arxiv: 2605.02420 · v1 · submitted 2026-05-04 · 🧮 math.PR

Large time behavior of critical marked Hawkes processes with heavy tailed marks and related branching particle systems

Anna Talarczyk This is my paper

Pith reviewed 2026-05-08 18:37 UTC · model grok-4.3

classification 🧮 math.PR

keywords critical marked Hawkes processesheavy tailed marksstable Lévy processesbranching particle systemsSkorokhod M1 topologylarge time asymptoticspoint processesspectrally positive stable laws

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The pith

In critical marked Hawkes processes with heavy-tailed marks, speeding up time causes the normalized counting process to converge to a spectrally positive stable Lévy process.

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

The paper studies the large-time behavior of critical marked Hawkes processes with a multiplicative kernel and nonnegative marks drawn independently from a distribution in the normal domain of attraction of a (1+β)-stable law where 0<β<1. It establishes that, for sufficiently small β, an appropriately normalized event-counting process converges in law to a spectrally positive 1/(1+β)-stable Lévy process when time is accelerated. The convergence takes place in the Skorokhod space of càdlàg paths under the M1 topology. The proof relies on a branching-particle-system representation of the Hawkes process, which also covers more general critical branching mechanisms attracted to stable laws. This complements earlier work on the same model for larger values of β and supplies a scaling limit for self-exciting point processes in the critical regime with infinite-variance marks.

Core claim

We show that, as the time is speeded up, if β is small enough then, the event counting process, appropriately normalized, converges to a spectrally positive 1/(1+β) stable Lévy process. The convergence holds in law in the Skorokhod space of càdlàg functions equipped with M1 topology. We also study a borderline case. The present paper complements the results of the same model studied for large β. We employ techniques involving a branching representation of marked Hawkes processes. This approach allows to study more general branching processes with branching mechanism in the normal domain of attraction of (1+β)-stable law.

What carries the argument

The branching representation of the marked Hawkes process, which recasts the counting process as a critical branching particle system whose offspring distribution lies in the normal domain of attraction of a (1+β)-stable law.

If this is right

The event-counting process admits a stable Lévy scaling limit rather than a Gaussian one when β is small.
Related critical branching particle systems with stable offspring laws obey the same limit.
The M1 topology accommodates the jumps of the limiting process, so the convergence remains valid even when paths are discontinuous.
A separate borderline case between the small-β and large-β regimes is handled by the same branching techniques.

Where Pith is reading between the lines

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

In applications such as seismology or financial contagion, critical self-exciting events with sufficiently heavy marks will exhibit stable rather than diffusive fluctuations at long horizons.
The choice of M1 topology indicates that the limit is robust under certain monotone time reparametrizations that preserve jump locations.
Numerical path simulations of the normalized counting process for increasing T could directly check whether the empirical stable index matches 1/(1+β).
The branching representation may extend to kernels that are no longer purely multiplicative, provided the offspring law remains in the stable domain of attraction.

Load-bearing premise

The marks are drawn independently and identically from a distribution whose tails place it in the normal domain of attraction of a (1+β)-stable law, and the mean number of offspring per event is exactly one.

What would settle it

Simulate many realizations of the marked Hawkes process to a large horizon T, apply the paper's normalization to the counting process at sped-up times, and compare the finite-dimensional distributions or path measures to those of the target 1/(1+β)-stable Lévy process; systematic mismatch in jump sizes or tail probabilities would falsify the claimed convergence.

read the original abstract

We study large time behavior of critical marked Hawkes processes and related branching particle systems. In case of marked Hawkes processes we assume that the kernel function has multiplicative form and the marks corresponding to the events are nonnegative and are assigned independently from a common distribution. This distribution is in the normal domain of attraction of a $(1+\beta)$-stable law with $0<\beta<1$. Moreover, we assume that the mean number of events triggered by a single event is equal to $1$ (criticality). We show that, as the time is speeded up, if $\beta$ is small enough then, the event counting process, appropriately normalized, converges to a spectrally positive $1/(1+\beta)$ stable L\'evy process. The convergence holds in law in the Skorokhod space of c\`adl\`ag functions equipped with $M_1$ topology. We also study a borderline case. The present paper complements the results of [A.Talarczyk:``A generalized central limit theorem for critical marked Hawkes processes'', arXiv:2504.11612], where the same model was studied in case of ``large'' $\beta$. We employ techniques involving a branching representation of marked Hawkes processes. This approach allows to study more general branching processes with branching mechanism in the normal domain of attraction of $(1+\beta)$-stable law.

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 extends the author's earlier large-β results to small β for critical marked Hawkes processes, getting convergence to a spectrally positive stable Lévy process in M1 topology via the branching representation.

read the letter

The main new piece is the small-β regime. The prior paper handled large β with one normalization; here the author switches the scaling and uses M1 topology to close the limit when β is small enough. The branching particle system representation turns the Hawkes counting process into a critical branching process whose offspring law sits in the normal domain of attraction of a (1+β)-stable distribution, and standard limit theorems for those processes give the Lévy limit. That reduction is clean and lets the argument cover the multiplicative kernel and independent marks without extra conditions. The borderline case is treated separately, which keeps the main theorem from stretching too far. The assumptions line up with what is already standard for this model, and the stress-test note confirms there is no hidden circularity or moment gap that breaks the M1 convergence. The only real limitation is the technical cutoff on β, which is needed for the moment estimates but is stated explicitly. Nothing in the abstract or setup suggests the proof relies on fitted parameters or self-referential limits. This is narrow but technically solid work on scaling limits for point processes. Specialists who already follow Hawkes processes, stable branching limits, or functional convergence in Skorokhod space will find the extension useful and the method reproducible. It is worth sending to a serious referee; the claim is precise, the approach is standard but applied in a new range, and the paper fills a clear gap left by the earlier work.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a functional limit theorem for critical marked Hawkes processes with multiplicative kernels and nonnegative marks whose distribution lies in the normal domain of attraction of a (1+β)-stable law (0<β<1). For sufficiently small β the normalized counting process converges in law to a spectrally positive 1/(1+β)-stable Lévy process in the Skorokhod space equipped with the M1 topology. The proof relies on the branching representation of the process; a borderline case is treated separately. The result complements the author's earlier work for larger β and extends to related branching particle systems.

Significance. If the central convergence holds, the paper supplies the missing small-β regime and thereby gives a complete description of the large-time scaling limits for this class of critical Hawkes processes under heavy-tailed marks. The branching-representation approach is a strength, as it permits general offspring distributions in the stable domain of attraction rather than restricting to specific kernels. The M1-topology convergence is technically nontrivial and connects the model to stable Lévy processes in a falsifiable way.

minor comments (2)

[Abstract] Abstract: the borderline-case result is announced but not stated; a one-sentence description of the limit obtained there would help the reader.
[Introduction] The title mentions related branching particle systems, yet the abstract focuses almost exclusively on the Hawkes process; a short paragraph clarifying the precise statement for the particle system would improve balance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The report correctly identifies the main result as completing the small-β regime for the scaling limits of critical marked Hawkes processes with heavy-tailed marks, using the branching representation to handle general offspring distributions in the stable domain of attraction. We appreciate the recognition that the M1-topology convergence is technically nontrivial and that the work complements our earlier paper for larger β.

Circularity Check

0 steps flagged

Minor self-citation to complementary case; derivation otherwise self-contained

full rationale

The central result follows from the standard branching representation of multiplicative-kernel marked Hawkes processes combined with known limit theorems for critical branching processes whose offspring law lies in the normal domain of attraction of a (1+β)-stable distribution. The single self-citation to the author's prior arXiv:2504.11612 paper addresses only the complementary large-β regime and supplies no load-bearing step for the small-β M1 convergence proved here. No equation reduces by construction to a fitted parameter, no ansatz is imported via citation, and the finite-dimensional-to-functional passage is handled by direct moment estimates under the stated small-β restriction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on standard properties of stable laws and branching processes; no new entities or fitted parameters are introduced in the abstract.

axioms (3)

domain assumption Marks are i.i.d. nonnegative random variables in the normal domain of attraction of a (1+β)-stable law
Stated explicitly in the abstract as the key distributional assumption.
domain assumption The process is critical: mean number of offspring per event equals 1
Required for the large-time scaling to be non-degenerate.
domain assumption Kernel has multiplicative form
Allows the branching representation to hold.

pith-pipeline@v0.9.0 · 5543 in / 1455 out tokens · 47772 ms · 2026-05-08T18:37:03.970846+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost/FunctionalEquation.lean (J-cost uniqueness) — RS cost J=½(x+x⁻¹)−1 is unrelated to the (1+β)-stable generating-function family used here washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

G_β(s) = s + (1−s)^{1+β}/(1+β) ... distribution of the number of offspring of a single particle is in the normal domain of attraction of (1+β)-stable law.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 1 canonical work pages · 1 internal anchor

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A. G. Hawkes. Hawkes processes and their applications to finance: a review, Quant. Finance 18(2018), no. 2, 193–198

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Hawkes, D

A.G. Hawkes, D. Oakes. A cluster process representation of a self-exciting process.J. Appl. Probability11 (1974), 493–503

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U. Horst,W. Xu. Functional limit theorems for marked Hawkes point measures. Stochastic Process. Appl. 134 (2021), 94–131

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Jaisson and M

T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes,Ann. Appl. Probab.25(2015), no. 2, 600–631

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Jaisson and M

T. Jaisson and M. Rosenbaum, Rough fractional diffusions as scaling limits of nearly un- stable heavy tailed Hawkes processes,Ann. Appl. Probab.26(2016), no. 5, 2860–2882

2016
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work page internal anchor Pith review Pith/arXiv arXiv 2025
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Whitt.Stochastic-Process Limits.Springer Series in Operations Research

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[1] [1]

Bacry, S

E. Bacry, S. Delattre, M. Hoffmann. Some limit theorems for Hawkes processes and appli- cation to financial statistics.Stochastic Process. Appl.123 (2013), no. 7, 2475–2499

2013

[2] [2]

Bingham, C.M

N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular variation. Encyclopedia Math. Appl., 27 Cambridge University Press, Cambridge, 1989. xx+494 pp

1989

[3] [3]

Daley, D

D.J. Daley, D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Ele- mentary theory and methods. Second edition Probab. Appl. (N. Y.) Springer-Verlag, New York, 2003

2003

[4] [4]

Durrett, Probability: theory and examples

R. Durrett, Probability: theory and examples. Fourth edition. Camb. Ser. Stat. Probab. Math., 31 Cambridge University Press, Cambridge, 2010

2010

[5] [5]

A.G. Hawkes. Point spectra of some mutually exciting point processes.J. Roy. Statist. Soc. Ser. B 33, 438–443, 1971. 23

1971

[6] [6]

A.G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika58 (1971), 83–90

1971

[7] [7]

A. G. Hawkes. Hawkes processes and their applications to finance: a review, Quant. Finance 18(2018), no. 2, 193–198

2018

[8] [8]

Hawkes, D

A.G. Hawkes, D. Oakes. A cluster process representation of a self-exciting process.J. Appl. Probability11 (1974), 493–503

1974

[9] [9]

U. Horst,W. Xu. Functional limit theorems for marked Hawkes point measures. Stochastic Process. Appl. 134 (2021), 94–131

2021

[10] [10]

Horst, W

U. Horst, W. Xu. Functional limit theorems for Hawkes processes.Probab. Theory Relat. Fields194(2026), no. 1-2, 917–996

2026

[11] [11]

Jaisson and M

T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes,Ann. Appl. Probab.25(2015), no. 2, 600–631

2015

[12] [12]

Jaisson and M

T. Jaisson and M. Rosenbaum, Rough fractional diffusions as scaling limits of nearly un- stable heavy tailed Hawkes processes,Ann. Appl. Probab.26(2016), no. 5, 2860–2882

2016

[13] [13]

Kern, Skorokhod topologies.Math Semesterber71(2024), 1–18

J. Kern, Skorokhod topologies.Math Semesterber71(2024), 1–18

2024

[14] [14]

P. J. Laub, Y. Lee, P.K. Pollett, T. Taimre, Hawkes models and their applications,Annu. Rev. Stat. Appl.12(2025), 233–258

2025

[15] [15]

In: Jiao, Y

Z.Li, Continuous-State Branching Processes with Immigration. In: Jiao, Y. (eds)From Probability to Finance.Mathematical Lectures from Peking University. Springer, Singapore. (2020)

2020

[16] [16]

A generalized central limit theorem for critical marked Hawkes processes

A. Talarczyk, A generalized central limit theorem for critical marked Hawkes processes, 2025, preprint arXiv:2504.11612 v3

work page internal anchor Pith review Pith/arXiv arXiv 2025

[17] [17]

Whitt.Stochastic-Process Limits.Springer Series in Operations Research

W. Whitt.Stochastic-Process Limits.Springer Series in Operations Research. Springer, New York, 2002 24

2002