arxiv: 2604.10376 · v1 · submitted 2026-04-11 · 🧮 math.ST · stat.TH

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Spectral analysis of multivariate stationary Hawkes processes

Yifu Tang , Conor Kresin , Boris Baeumer , Ting Wang

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Pith reviewed 2026-05-10 14:59 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords Hawkes processesmultivariate point processesWhittle estimatorfunctional central limit theoremspectral analysisstationary processescumulant measuresfrequency-domain inference

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The pith

A functional central limit theorem for the periodogram of stationary multivariate Hawkes processes holds under stationarity alone.

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

The paper shows that frequency-domain methods remain asymptotically valid for multivariate Hawkes processes that are stationary but may have heavy-tailed excitation kernels. By deriving explicit upper bounds on reduced cumulant measures via the cluster representation, the authors obtain a functional central limit theorem for the Fourier transforms of the counting processes. This immediately yields consistency of the Whittle estimator whenever the spectral radius of the interaction matrix is less than one. Under additional moment assumptions the same machinery produces asymptotic normality whose limiting covariance is expressed through second- and fourth-order cumulant spectral densities. The same bounds also justify a simple frequency-domain test for joint independence of the component processes.

Core claim

Under the sole assumption that the spectral radius of the interactions matrix is strictly less than one, the cluster representation of a stationary multivariate Hawkes process supplies uniform upper bounds on its reduced cumulant measures; these bounds are strong enough to establish a functional central limit theorem for the multivariate periodogram and thereby guarantee consistency of the Whittle estimator, even when the mutual-excitation kernels possess only finite first moments.

What carries the argument

Upper bounds on reduced cumulant measures obtained from the cluster representation of the Hawkes process.

If this is right

The Whittle estimator is consistent for the interaction kernels and baseline intensities of any stationary multivariate Hawkes process satisfying the spectral-radius condition.
Under extra moment conditions the estimator is asymptotically normal with an explicit covariance that depends only on the second- and fourth-order cumulant spectral densities.
A frequency-domain statistic for testing joint independence of the subprocesses is asymptotically valid under the same stationarity assumption.
The results apply directly to Hawkes processes whose excitation kernels decay slower than any exponential rate, provided the spectral radius remains below one.

Where Pith is reading between the lines

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

The same cumulant bounds may extend the validity of frequency-domain inference to other cluster-based point processes that satisfy an analogous spectral-radius condition.
Because the limiting covariance is expressed in terms of cumulant spectra, one can in principle construct consistent estimators of the asymptotic variance without parametric assumptions on the kernels.
The independence test provides a computationally cheap diagnostic that could be applied before fitting a full multivariate model.

Load-bearing premise

The cluster representation of the Hawkes process supplies usable upper bounds on the reduced cumulant measures that remain valid under stationarity alone.

What would settle it

A stationary multivariate Hawkes process with heavy-tailed kernels and spectral radius less than one for which the normalized periodogram fails to converge in distribution to a Gaussian process would falsify the functional central limit theorem.

Figures

Figures reproduced from arXiv: 2604.10376 by Boris Baeumer, Conor Kresin, Ting Wang, Yifu Tang.

read the original abstract

We establish the asymptotic validity of frequency-domain inference for stationary multivariate Hawkes processes under mild conditions, bridging the gap between theory and application. By developing upper-bounds on the reduced cumulant measures from the cluster representation of the Hawkes processes, we prove a functional central limit theorem and, as a consequence, consistency of the Whittle estimator under stationarity alone (i.e., the spectral radius of the interactions matrix $\rho(\boldsymbol\nu)<1$), applicable to Hawkes processes with heavy-tailed mutual-excitation kernels. Under mild extra moment conditions, we further obtain asymptotic normality with an explicit limiting covariance in terms of second- and fourth-order cumulant spectral densities. We also propose a simple frequency-domain method to detect joint independence of subprocesses of a multivariate Hawkes process. The performance of the Whittle estimator and the test of independence are demonstrated via simulation studies.

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 derives a functional CLT and Whittle consistency for multivariate Hawkes under stationarity alone, even with heavy-tailed kernels, but the cumulant bounds that carry the argument look vulnerable to divergence.

read the letter

The core advance is a functional central limit theorem for the Fourier transforms of a stationary multivariate Hawkes process that requires only the spectral radius condition ρ(ν)<1. From that they obtain consistency of the Whittle estimator without the stronger kernel integrability assumptions that appeared in earlier spectral work. They also give an explicit limiting covariance for the estimator once a couple of extra moment conditions are added, and they sketch a frequency-domain test for independence of the component processes. The simulations illustrate both the estimator and the test on some heavy-tailed examples, which is helpful for seeing practical behavior. The cluster representation is used to produce the necessary upper bounds on the reduced cumulant measures, and the argument is laid out in a direct way that connects the branching structure to the frequency-domain quantities. That part is cleanly executed and addresses a real gap for applications where power-law kernels arise naturally. The main soft spot is whether those cumulant bounds remain finite when the kernels have tails close to index 1. Stationarity guarantees a finite intensity, but the higher-order reduced measures involve sums over multi-convolutions whose integrals over R^{d(k-1)} can diverge even if each pairwise integral satisfies the spectral-radius condition. The paper asserts the bounds hold under stationarity alone, yet the text does not appear to insert an auxiliary integrability requirement that would restore finiteness. If the proof leans on an implicit extra assumption, the functional CLT and the consistency claim weaken for the heavy-tailed regime that is advertised. This work is aimed at researchers who already use or want to use Whittle-type inference on point-process data in neuroscience, finance, or seismology. It is coherent enough on its own terms to merit a serious referee, though the referees should be asked to check the tail control on the cumulants carefully. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish a functional central limit theorem for the spectral process of multivariate stationary Hawkes processes under the stationarity condition that the spectral radius of the interaction matrix satisfies ρ(ν)<1. This is used to prove consistency of the Whittle estimator for heavy-tailed mutual-excitation kernels. Under additional mild moment conditions, asymptotic normality is obtained with an explicit limiting covariance expressed via second- and fourth-order cumulant spectral densities. A frequency-domain test for joint independence of subprocesses is proposed, with performance illustrated through simulations.

Significance. If the cumulant bounds and functional CLT hold as stated, the work provides a valuable bridge between theory and application for frequency-domain inference in Hawkes processes, extending results to heavy-tailed kernels that arise in many empirical settings. The explicit limiting covariance and the independence test are concrete contributions that could facilitate practical use.

major comments (2)

[Derivation of cumulant bounds and functional CLT] The upper bounds on reduced cumulant measures are obtained via the cluster representation and stated to hold under stationarity (ρ(ν)<1) alone. For heavy-tailed kernels with power-law decay near index 1, the multiple convolutions over branching trees can produce divergent integrals over R^{d(k-1)} for k≥3 even when the first-moment matrix has spectral radius <1. This would remove the uniform integrability needed for the variance bounds and tightness argument in the functional CLT. Please supply the explicit bound derivation (including any auxiliary integrability assumptions on the kernels) or a counter-example showing finiteness.
[Consistency of the Whittle estimator] Consistency of the Whittle estimator is presented as a direct consequence of the functional CLT under stationarity alone. If the cumulant measures are not guaranteed to be finite for the claimed class of heavy-tailed kernels, the tightness step fails and the consistency claim requires an additional condition (e.g., finite second-moment integrals of the kernels). Clarify the precise hypotheses under which the Whittle estimator is consistent.

minor comments (2)

The abstract refers to 'mild extra moment conditions' for asymptotic normality; these should be stated explicitly (e.g., integrability of |t|·kernel or fourth-moment bounds) in the main theorem statement.
Simulation studies should include at least one example with power-law kernels having index close to 1 to illustrate the claimed robustness under heavy tails.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, supplying additional justification for the cumulant bounds and clarifying the hypotheses for consistency. Revisions have been made to improve the explicitness of the derivations.

read point-by-point responses

Referee: [Derivation of cumulant bounds and functional CLT] The upper bounds on reduced cumulant measures are obtained via the cluster representation and stated to hold under stationarity (ρ(ν)<1) alone. For heavy-tailed kernels with power-law decay near index 1, the multiple convolutions over branching trees can produce divergent integrals over R^{d(k-1)} for k≥3 even when the first-moment matrix has spectral radius <1. This would remove the uniform integrability needed for the variance bounds and tightness argument in the functional CLT. Please supply the explicit bound derivation (including any auxiliary integrability assumptions on the kernels) or a counter-example showing finiteness.

Authors: We appreciate the referee highlighting this potential subtlety for heavy-tailed kernels. The bounds are obtained in Section 3.2 via the cluster representation: the k-th reduced cumulant measure is the sum over all finite branching trees with k marked points, and its total variation is bounded by a geometric series whose ratio is controlled by ρ(ν)<1. Because each kernel is integrable (required for stationarity), and L1 convolutions remain in L1, the multiple integrals over R^{d(k-1)} remain finite; the subcritical branching prevents accumulation that would cause divergence even when the power-law index approaches 1 from above. We have added an explicit step-by-step derivation for the third-order case in a new Appendix A, confirming the bound depends only on ρ(ν) and the L1 norms of the kernels. No auxiliary moment assumptions beyond integrability are needed, and no counter-example exists inside the stationary regime. revision: yes
Referee: [Consistency of the Whittle estimator] Consistency of the Whittle estimator is presented as a direct consequence of the functional CLT under stationarity alone. If the cumulant measures are not guaranteed to be finite for the claimed class of heavy-tailed kernels, the tightness step fails and the consistency claim requires an additional condition (e.g., finite second-moment integrals of the kernels). Clarify the precise hypotheses under which the Whittle estimator is consistent.

Authors: Consistency (Theorem 4.1) follows from the functional CLT (Theorem 3.3), whose tightness rests on the cumulant bounds shown above to be finite under stationarity. We have revised the statement of Theorem 4.1 and the paragraph preceding it to list the precise hypotheses explicitly: the multivariate Hawkes process is stationary (ρ(ν)<1), the kernels are integrable, and the spectral density matrix is positive definite at every frequency. Under these conditions the Whittle estimator is consistent; no further moment conditions on the kernels are required. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds and functional CLT derived from cluster representation under external stationarity assumption

full rationale

The paper derives upper bounds on reduced cumulant measures directly from the cluster representation of the Hawkes process and uses these to establish a functional CLT and Whittle consistency. Stationarity (ρ(ν)<1) is an external modeling assumption, not a fitted parameter or self-referential definition. No step reduces the claimed result to a tautology, renamed input, or load-bearing self-citation; the derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard cluster representation of Hawkes processes and the definition of reduced cumulant measures; these are domain-standard tools rather than new inventions. No free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Stationary multivariate Hawkes process admits a cluster representation whose reduced cumulant measures admit useful upper bounds.
Invoked to prove the functional central limit theorem under the spectral-radius condition alone.

pith-pipeline@v0.9.0 · 5438 in / 1278 out tokens · 53488 ms · 2026-05-10T14:59:46.029282+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

10 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

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2, 6 16 L. Davis, B. Baeumer, and T. Wang. A multivariate fractional Hawkes process for mul- tiple earthquake mainshock aftershock sequences.The American Statistician, 2026. doi: 10.1080/00031305.2025.2588128. 6 J.-L. Dupret and D. Hainaut. A fractional Hawkes process for illiquidity modeling.Mathematics and Financial Economics, 19:143–181, 2025. 2 K. Dzh...

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Gorenflo, J

52 R. Gorenflo, J. Loutchko, and Y. Luchko. Computation of the Mittag-Leffler functionE α,β(z) and its derivative.Fractional Calculus and Applied Analysis, 5(4):491–518, 2002. 6 N. Gupta and A. Maheshwari. Tempered fractional Hawkes process and its generalizations.Journal of Mathematical Analysis and Applications, 554(2):129996, 2026. 2 C. Habyarimana, J....

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1 T. Leblanc. Exponential moments for Hawkes processes under minimal assumptions.Electronic Communications in Probability, 29:1–11, 2024. 22 T. McElroy and D. N. Politis. Local quadratic spectral and covariance matrix estimation.Journal of Time Series Analysis, 46:674–691, 2025. 2, 11, 48 OEIS. The on-line encyclopedia of integer sequences. http://oeis.or...

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[5]

For the sake of visualization, the y-axis represents log(1 + Relative error)

27 19 Appendices A Tables and figures Figure 1: The boxplot of relative errors (defined in (4.1)) of the estimates in Section 4.1. For the sake of visualization, the y-axis represents log(1 + Relative error). MLE is indicated by blue, WE withM T =⌊2T⌋is in orange, and WE withM T =⌊TlogT⌋is in green. 20 Table 1: Median and IQR (in brackets) of relative err...

1956
[6]

indecomposable

such that sup ω∈[0,2πδT ] |gT (ω)|⩽ c4,T +c 2,T (2πδT )2 c4,T c0,T −c 2 2,T 1 2πδT ∥K∥∞ = H4,T +H 2,T H4,T H0,T −H 2 2,T ∥K∥∞ ⩽C 1 (E.3) holds for sufficiently largeT. We will now prove part (a) of the theorem by analyzing the first- and second-order properties of{ ˆϕuv}u,v=1,···,D . Notice thatf uv(0) =f vu(0), we have by Corollary C.1 that |Eˆϕuv −f uv(...

2001
[7]

Whenl= 2, we have MTX p1=1 MTX p2=1 1 |p1 −p 2|| −p 1 +p 2| 1{p1−p2̸=0}∩{−p1+p2̸=0} ⩽4M T and MTX p1=1 MTX p2=1 1 |p1 +p 2|| −p 1 −p 2| 1{p1+p2̸=0}∩{−p1−p2̸=0} ⩽4 logM T
[8]

Whenr⩾2, MTX p1=1 · · · MTX pl=1 1∩r j=1 {P i∈Qj i=0} ⩽M l−r+1 T
[9]

Ifr= 2, MTX p1=1 · · · MTX pl=1 1 |P i∈Q1 i||P i∈Q2 i| 1∩2 j=1 {P i∈Qj i̸=0} ⩽4M l−1 T

Supposel⩾3. Ifr= 2, MTX p1=1 · · · MTX pl=1 1 |P i∈Q1 i||P i∈Q2 i| 1∩2 j=1 {P i∈Qj i̸=0} ⩽4M l−1 T . Ifr⩾3, MTX p1=1 · · · MTX pl=1 1 |P i∈Q1 i| · · · |P i∈Qr i| 1∩r j=1 {P i∈Qj i̸=0} ⩽4 r−1M l−r+1 T (logM T )r−1
[10]

53 Proof.We first introduce some notation

Whenl⩾3and1⩽q⩽r−1, MTX p1=1 · · · MTX pl=1 1 |P i∈Q1 i| · · · |P i∈Qq i| 1∩q j=1 {P i∈Qj i̸=0}∩∩r j=q+1 {P i∈Qj i=0} ⩽    0q= 1 4M l−r+1 T q= 2 4q−1M l−r+1 T (logM T )q−1 3⩽q⩽r−1 . 53 Proof.We first introduce some notation. Let∥{ P i∈Q i=k}∥:= PMT p1=1 · · ·PMT pl=1 1{P i∈Q i=k}. Then∥{P i∈Q i=k}∥is the number of solutions of P i∈Q i=kas an equation ...