A generalized central limit theorem for critical marked Hawkes processes

Anna Talarczyk

arxiv: 2504.11612 · v3 · submitted 2025-04-15 · 🧮 math.PR

A generalized central limit theorem for critical marked Hawkes processes

Anna Talarczyk This is my paper

Pith reviewed 2026-05-22 19:43 UTC · model grok-4.3

classification 🧮 math.PR

keywords Hawkes processescentral limit theoremmarked point processesstable processescriticalityself-exciting systemstempered distributions

0 comments

The pith

Critical marked Hawkes processes converge in law to stable or Gaussian processes depending on mark distributions.

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

This paper proves a central limit theorem for critical marked Hawkes processes with i.i.d. nonnegative marks. When marks have heavy tails, the limit of the normalized empirical measure is a stable process with dependent increments. When marks have finite variance, the limit is the Gaussian process known from unmarked critical Hawkes processes. The proof uses convergence in the space of tempered distributions and applies a new robust method for self-exciting systems. This matters because it extends understanding of clustering in point processes under critical conditions where standard laws fail.

Core claim

We prove convergence in law in the space of tempered distributions of the normalized empirical measure corresponding to the times of events. In case when the distribution of marks is heavy tailed, the limit process is a stable process with dependent increments, while in case of finite variance, the limit process is the same Gaussian process as for the non marked Hawkes process. We also study convergence in law in the Skorokhod space of the normalized event counting process as the time is speeded up. We develop a new, robust method that may be applied to other self-exciting systems generalizing Hawkes processes. For example we consider a non marked self-exciting system where the number of exc

What carries the argument

The multiplicative kernel with criticality (mean of one offspring per event) and heavy-tailed base intensity, which together allow the normalized empirical measure to converge to the described limits.

If this is right

Convergence holds in the Skorokhod space for the sped-up normalized counting process.
Heavy-tailed mark distributions yield stable limits with dependent increments.
Finite-variance marks recover the unmarked Gaussian limit.
The method generalizes to non-marked self-exciting systems with heavy-tailed excitations.

Where Pith is reading between the lines

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

The result suggests that marks can induce dependence even in stable regime limits for critical processes.
This framework could be tested on real data from fields like finance or seismology where Hawkes-like models are used.
Extensions to other self-exciting point processes with similar criticality might follow similar convergence patterns.

Load-bearing premise

The kernel function must be of multiplicative form, the process must be exactly critical with average one future event per event, and the base intensity function must be heavy tailed.

What would settle it

Observing that the normalized empirical measure of event times from a simulated critical marked Hawkes process does not converge in law to the predicted stable or Gaussian process in the space of tempered distributions would falsify the central claim.

read the original abstract

We prove a central limit type theorem for critical marked Hawkes processes. We study the case where the marks are i.i.d. with nonnegative values and their common distribution is either heavy tailed or has finite variance. The kernel function is of a multiplicative form and the mean number of future events triggered by a single event is $1$ (criticality). We also assume that the base intensity function is heavy tailed. We prove convergence in law in the space of tempered distributions of the normalized empirical measure corresponding to the times of events. We also study convergence in law in the Skorokhod space of the normalized event counting process as the time is speeded up. In case when the distribution of marks is heavy tailed, the limit process is a stable process with dependent increments, while in case of finite variance, the limit process is the same Gaussian process as for the non marked Hawkes process. We develop a new, robust method that may be applied to other self-exciting systems generalizing Hawkes processes. For example we consider a non marked self-exciting system where the number of excitations caused by single event is heavy tailed.

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 proves a CLT for the normalized empirical measure of critical marked Hawkes processes, yielding a stable limit with dependent increments under heavy-tailed marks and a Gaussian one otherwise, but the multiplicative kernel is doing the heavy lifting.

read the letter

The core result is a convergence in law for the normalized point measure of event times in a critical marked Hawkes process, where the base intensity is heavy-tailed. When marks have heavy tails the limit is a stable process with dependent increments; finite-variance marks recover the usual Gaussian limit from the unmarked case. They also sketch an extension to a non-marked self-exciting system whose offspring distribution is heavy-tailed. That is the actual new piece: the marked critical regime with the stated tail conditions, plus a method they claim works for broader self-exciting models. The abstract is clear on the assumptions, including the multiplicative kernel form K(t,m) = k(t)·m and criticality (mean offspring = 1). The approach looks technically standard—characteristic functionals and convergence in tempered distributions—but the heavy-tail handling and the identification of the limit process are the parts that need the full proof to check. The multiplicative kernel lets the mark distribution factor out of the compensator; without it the cross terms would not be controlled by the base-intensity tail alone, so the “generalized” framing is narrower than it first sounds. The paper is explicit about the kernel restriction, which is good, but it does mean the result is not yet a black-box tool for arbitrary mark-dependent kernels. This is for people working on limit theorems for point processes and Hawkes-type models in probability. A reader who already knows the unmarked critical case will see the extension cleanly and can judge whether the method travels to their own variant. The work is coherent on its own terms and cites the relevant prior CLTs, so it is worth sending to a referee who can verify the technical steps in sections 3–4. I would not cite it myself yet, but it belongs in the review pipeline.

Referee Report

2 major / 2 minor

Summary. The paper proves a central limit type theorem for critical marked Hawkes processes with i.i.d. nonnegative marks that are either heavy-tailed or have finite variance. Under a multiplicative kernel form and heavy-tailed base intensity, it establishes convergence in law of the normalized empirical measure of event times in the space of tempered distributions, yielding a stable process with dependent increments for heavy-tailed marks and the same Gaussian process as the unmarked case for finite-variance marks. Convergence of the normalized counting process is also studied in Skorokhod space, and the method is illustrated on a non-marked self-exciting system with heavy-tailed excitations.

Significance. If the proofs are complete, the work supplies a robust new technique for limit theorems in self-exciting point processes that accommodates heavy tails in both marks and base intensity. The explicit identification of the limit objects and the extension to other systems constitute a clear contribution to the theory of Hawkes processes and their generalizations.

major comments (2)

[Abstract and §2] Abstract and §2 (kernel assumption): The multiplicative form K(t,m)=k(t)·m is load-bearing for decoupling the mark distribution from the intensity recursion and for controlling the characteristic functional that identifies the stable limit with dependent increments. The paper frames the result as a 'generalized' CLT for marked Hawkes processes, yet the derivation does not address whether the same limit identification survives for kernels that depend jointly on time and mark; without this, the extra cross terms may not be dominated by the heavy-tailed base intensity alone.
[Main convergence statement (likely Theorem 3.1 or §3)] Main convergence statement (likely Theorem 3.1 or §3): The claim that the finite-variance mark case recovers exactly the same Gaussian process as the non-marked Hawkes process requires explicit verification that the mark contribution vanishes in the limit after normalization; the interaction between the finite-variance mark distribution and the heavy-tailed base intensity needs sharper bounds to confirm this reduction.

minor comments (2)

[Notation and preliminaries] The definition of the space of tempered distributions and the topology used for convergence could include a short reference to a standard text for accessibility.
[Proofs] A few indices in the characteristic-functional calculations in the proof section are ambiguous and would benefit from explicit cross-references to earlier lemmas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing to make revisions where they strengthen the presentation and clarify the scope of our results.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (kernel assumption): The multiplicative form K(t,m)=k(t)·m is load-bearing for decoupling the mark distribution from the intensity recursion and for controlling the characteristic functional that identifies the stable limit with dependent increments. The paper frames the result as a 'generalized' CLT for marked Hawkes processes, yet the derivation does not address whether the same limit identification survives for kernels that depend jointly on time and mark; without this, the extra cross terms may not be dominated by the heavy-tailed base intensity alone.

Authors: We agree that the multiplicative kernel form K(t,m)=k(t)·m is essential to our decoupling argument and to controlling the characteristic functional for the stable limit. The manuscript explicitly assumes this form in Section 2 and uses it throughout the proofs. The term 'generalized' in the title and abstract refers to the extension from unmarked to marked critical Hawkes processes (with either heavy-tailed or finite-variance marks) under this standard multiplicative structure, together with the new method's applicability to other self-exciting systems. We do not claim the identical limit identification holds for arbitrary joint (t,m)-dependent kernels, where additional cross terms would indeed require separate control. To prevent any misinterpretation, we will revise the abstract and the opening of Section 2 to state the kernel assumption more prominently and to note that extensions beyond the multiplicative case lie outside the present scope. revision: yes
Referee: [Main convergence statement (likely Theorem 3.1 or §3)] Main convergence statement (likely Theorem 3.1 or §3): The claim that the finite-variance mark case recovers exactly the same Gaussian process as the non-marked Hawkes process requires explicit verification that the mark contribution vanishes in the limit after normalization; the interaction between the finite-variance mark distribution and the heavy-tailed base intensity needs sharper bounds to confirm this reduction.

Authors: We thank the referee for highlighting this point. Our current argument shows that, under finite variance of the marks, their contribution is o(1) after the appropriate normalization and therefore disappears in the limit, yielding the same Gaussian process obtained in the unmarked case. Nevertheless, we acknowledge that sharper quantitative bounds on the interaction between the finite-variance marks and the heavy-tailed base intensity would make the reduction fully explicit. We will add these estimates in the revised version of the proof of the finite-variance case (around the relevant theorem in Section 3) to confirm that the mark terms vanish uniformly in the limit. revision: yes

Circularity Check

0 steps flagged

No circularity detected in the proof of the generalized CLT for critical marked Hawkes processes

full rationale

The paper is a self-contained mathematical proof relying on standard techniques from probability theory for point processes and convergence in tempered distributions or Skorokhod space. Explicit assumptions (multiplicative kernel form, criticality with mean offspring=1, heavy-tailed base intensity) are stated upfront and used to derive the limit processes (stable with dependent increments for heavy-tailed marks; Gaussian for finite variance). No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the derivation does not rename known results or smuggle ansatzes via prior work. The result holds under the stated conditions without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard mathematical background for point processes and weak convergence; no free parameters are fitted, no new entities postulated, and criticality is an explicit modeling assumption rather than a derived quantity.

axioms (1)

standard math Standard results on weak convergence in Skorokhod space and in the space of tempered distributions for point processes.
Used to establish the stated convergence in law.

pith-pipeline@v0.9.0 · 5719 in / 1266 out tokens · 76439 ms · 2026-05-22T19:43:29.061432+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.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

kernel function of the form ϕ(s,x)=xφ(s)... mean number of future events triggered by a single event is 1 (criticality)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

branching particle system... resolvent R(s)... potential G(α)

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Large time behavior of critical marked Hawkes processes with heavy tailed marks and related branching particle systems
math.PR 2026-05 unverdicted novelty 6.0

For small β, the normalized event counting process of critical marked Hawkes processes with (1+β)-stable marks converges in law to a spectrally positive 1/(1+β)-stable Lévy process in Skorokhod M1 topology.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 1 Pith paper

[1]

Bacry, S

E. Bacry, S. Delattre, M. Hoffmann. Some limit theorems for Hawkes processes and ap- plication to financial statistics. Stochastic Process. Appl. 123 (2013), no. 7, 2475–2499. 32

work page 2013
[2]

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work page 1968
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P. P. Billingsley, Convergence of probability measures, second edition, Wiley Series in Prob- ability and Statistics: Probability and Statistics A Wiley-Interscience Publication, Wiley, New York, 1999

work page 1999
[4]

Bojdecki, L

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 (2007), no. 2, 500–527

work page 2007
[5]

Bojdecki, L

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. Occupation time fluctuations of an infinite- variance branching system in large dimensions. Bernoulli 13 (2007), no. 1, 20–39

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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

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Dawson, L.G

D.A. Dawson, L.G. Gorostiza, A. Wakolbinger. Degrees of transience and recurrence and hierarchical random walks. Potential Anal. 22 (2005), no. 4, 305–350

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S. N. Ethier and T. G. Kurtz, Markov processes, Wiley Series in Probability and Mathem- atical Statistics: Probability and Mathematical Statistics, Wiley, New York, 1986

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Hardiman, N

S.J. Hardiman, N. Bercot, J.P. Bouchaud, Critical reflexivity in financial markets: a Hawkes process analysis. Eur. Phys. J. B (2013) 86, 442

work page 2013
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A.G. Hawkes. Point spectra of some mutually exciting point processes. J. Roy. Statist. Soc. Ser. B 33, 438–443, 1971

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A.G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biomet- rika 58 (1971), 83–90

work page 1971
[12]

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

work page 2018
[13]

Hawkes, D

A.G. Hawkes, D. Oakes. A cluster process representation of a self-exciting process. J. Appl. Probability 11 (1974), 493–503

work page 1974
[14]

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

work page 2021
[15]

Horst, W

U. Horst, W. Xu. Functional limit theorems for Hawkes processes. Probab. Theory Relat. Fields (2024). https://doi.org/10.1007/s00440-024-01348-3

work page doi:10.1007/s00440-024-01348-3 2024
[16]

Horst, W

U. Horst, W. Xu and R. Zhang, , Convergence of heavy-tailed Hawkes processes and the microstructure of rough volatility,” arXiv:2312.08784, 2023

work page arXiv 2023
[17]

I. Iscoe. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71 (1986), no. 1, 85–116

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K. Itˆ o. Foundations of stochastic differential equations in infinite-dimensional spaces. CBMS-NSF Regional Conf. Ser. in Appl. Math., 47, SIAM, Philadelphia, 1984

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[19]

Jaisson and M

T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes, Ann. Appl. Probab. 25 (2015), no. 2, 600–631. 33

work page 2015
[20]

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

work page 2016
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P. J. Laub, Y. Lee, P.K. Pollett, T. Taimre, Hawkes models and their applications, Annu. Rev. Stat. Appl. 12 (2025), 233–258

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S. Peszat, J. Zabczyk Stochastic partial differential equations with L´ evy noise. An evolution equation approach Encyclopedia Math. Appl., 113 Cambridge University Press, Cambridge, 2007

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and Taqqu, M.S., (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York

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[1] [1]

Bacry, S

E. Bacry, S. Delattre, M. Hoffmann. Some limit theorems for Hawkes processes and ap- plication to financial statistics. Stochastic Process. Appl. 123 (2013), no. 7, 2475–2499. 32

work page 2013

[2] [2]

Breiman, Probability

L. Breiman, Probability. Addison–Wesley, Reading, MA., 1968

work page 1968

[3] [3]

P. P. Billingsley, Convergence of probability measures, second edition, Wiley Series in Prob- ability and Statistics: Probability and Statistics A Wiley-Interscience Publication, Wiley, New York, 1999

work page 1999

[4] [4]

Bojdecki, L

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 (2007), no. 2, 500–527

work page 2007

[5] [5]

Bojdecki, L

T. Bojdecki, L. G. Gorostiza, and A. Talarczyk. Occupation time fluctuations of an infinite- variance branching system in large dimensions. Bernoulli 13 (2007), no. 1, 20–39

work page 2007

[6] [6]

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

work page 2003

[7] [7]

Dawson, L.G

D.A. Dawson, L.G. Gorostiza, A. Wakolbinger. Degrees of transience and recurrence and hierarchical random walks. Potential Anal. 22 (2005), no. 4, 305–350

work page 2005

[8] [8]

S. N. Ethier and T. G. Kurtz, Markov processes, Wiley Series in Probability and Mathem- atical Statistics: Probability and Mathematical Statistics, Wiley, New York, 1986

work page 1986

[9] [9]

Hardiman, N

S.J. Hardiman, N. Bercot, J.P. Bouchaud, Critical reflexivity in financial markets: a Hawkes process analysis. Eur. Phys. J. B (2013) 86, 442

work page 2013

[10] [10]

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

work page 1971

[11] [11]

A.G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biomet- rika 58 (1971), 83–90

work page 1971

[12] [12]

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

work page 2018

[13] [13]

Hawkes, D

A.G. Hawkes, D. Oakes. A cluster process representation of a self-exciting process. J. Appl. Probability 11 (1974), 493–503

work page 1974

[14] [14]

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

work page 2021

[15] [15]

Horst, W

U. Horst, W. Xu. Functional limit theorems for Hawkes processes. Probab. Theory Relat. Fields (2024). https://doi.org/10.1007/s00440-024-01348-3

work page doi:10.1007/s00440-024-01348-3 2024

[16] [16]

Horst, W

U. Horst, W. Xu and R. Zhang, , Convergence of heavy-tailed Hawkes processes and the microstructure of rough volatility,” arXiv:2312.08784, 2023

work page arXiv 2023

[17] [17]

I. Iscoe. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71 (1986), no. 1, 85–116

work page 1986

[18] [18]

K. Itˆ o. Foundations of stochastic differential equations in infinite-dimensional spaces. CBMS-NSF Regional Conf. Ser. in Appl. Math., 47, SIAM, Philadelphia, 1984

work page 1984

[19] [19]

Jaisson and M

T. Jaisson and M. Rosenbaum, Limit theorems for nearly unstable Hawkes processes, Ann. Appl. Probab. 25 (2015), no. 2, 600–631. 33

work page 2015

[20] [20]

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

work page 2016

[21] [21]

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

work page 2025

[22] [22]

Ogata, Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes, Journal of the American Statistical Association, Vol

Y. Ogata, Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes, Journal of the American Statistical Association, Vol. 83, No. 401 (Mar., 1988), 9–27

work page 1988

[23] [23]

Peszat, J

S. Peszat, J. Zabczyk Stochastic partial differential equations with L´ evy noise. An evolution equation approach Encyclopedia Math. Appl., 113 Cambridge University Press, Cambridge, 2007

work page 2007

[24] [24]

and Taqqu, M.S., (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York

Samorodnitsky, G. and Taqqu, M.S., (1994), Stable Non-Gaussian Random Processes, Chapman & Hall, New York. 34

work page 1994