{"id":"dd2fd56f-cea0-43b5-a454-e8f2e0cd604a","arxiv_id":"1908.06703","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The rescaled marked Hawkes point measure with immigration converges to a Gaussian white noise plus a Brownian-motion lifting, and the shot noise converges to a Brownian martingale.","lead":"This paper proves precise statistical laws for marked Hawkes processes, event models where each event can trigger more events and carries a label. The results describe the random fluctuations around the average behavior at large time scales, with application to toxins released by budding microbes in a host.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 4.1 Volterra representation is unverified and as printed is inconsistent with its use in (4.7); the theorem may hold, but the proof rests on an uncorrected identity.","rationale":"The reader's weakest-assumption identification is in the right place: Proposition 4.1 is indeed the hinge on which the functional CLTs depend, and its provenance from the unpublished preprint [39] is not accompanied by a verification of hypotheses. I do not find a reason to move away from the reader's CONDITIONAL verdict. However, the concern is more concrete than 'unverified external theorem': as printed, Eq. (4.6) contains a Volterra term with the terminal time t inside the integrand, while Eq. (4.7) and all later estimates integrate a causal convolution in s. This is either a typo that should be corrected or a genuine switch of representation inside the proof. The good news is that the scalar intensity equation (2.3) can likely be solved directly by the resolvent corresponding to φ_H, so a short self-contained derivation of Proposition 4.1 would settle the issue without relying on [39]. The detailed moment and tightness estimates in Section 4 are substantial independent support for the rest of the argument; this concern is about the input to those estimates, not about the estimates themselves. Recommendation: keep the paper CONDITIONAL, with the condition being a correct, self-contained statement and proof of Proposition 4.1, including resolution of the µ0(t) versus µ0(s) inconsistency.","tokens_in":34035,"tokens_out":12508,"duration_ms":120770,"concrete_test":"Independently derive Proposition 4.1 directly from Eq. (2.3): split N0 into its compensator plus the martingale measure, rewrite the original intensity equation as Z(t) = µ0(t) + (φ_H * Z)(t) + a source term, and solve this linear Volterra equation with the resolvent kernel R_H. This yields the exact form of the second term. If it is ∫_0^t R_H(t-s) µ0(s) ds, correct Eqs. (4.3) and (4.6) accordingly and confirm that (4.7)–(4.13) and Lemma 4.2 are unaffected; if any extra term or integrability condition appears, verify that Condition 3.2 implies it and that Lemma 4.2 still covers the new term. This check also settles whether the external theorem [39, Thm 2.2] is needed at all.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central CLT (Theorem 3.11) is proved through the stochastic Volterra representation Proposition 4.1 / Eq. (4.6), so the soundness of that representation is load-bearing. Two mutually reinforcing problems make this the weakest link. First, the representation is imported from the same authors' unpublished preprint [39], and the hypotheses of the quoted Theorem 2.2 are never checked against Condition 3.2; the reader cannot tell whether the identity is valid in this setting. Second, the equation as printed is internally inconsistent: Eq. (4.6) writes the second term as ∫_0^t R_H(t-s) µ0(t) ds, with the terminal time inside the integrand, whereas the subsequent integration in Eq. (4.7) uses ∫_0^{Tt} R_H(Tt-s) ds ∫_0^s µ0(r) dr, which corresponds to a causal Volterra term ∫_0^t R_H(t-s) µ0(s) ds. All estimates in Lemmas 4.2–4.4 and Propositions 3.9–3.10 rely on the causal form. If (4.3)/(4.6) are typographical, they should be corrected; if the terminal-time form is genuinely what [39] supplies, then the proof silently switches to a different representation and the CLT proof has a real gap. The theorem may be true, but the representation that carries the proof must be verified and stated consistently before the proof is complete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies marked Hawkes point measures with homogeneous immigration and establishes functional laws of large numbers and functional central limit theorems. Under moment and tail conditions (Condition 3.2) and the stability condition ‖φ_H‖_{L1}<1, the main result (Theorem 3.11) states that the rescaled point-measure error process √T \\bar N^T_{H,t}(A) converges weakly in D([0,∞),S(U)) to W_{H,t}(A)+σ_Z B_Z(t)ν_H(A), where W_H is a Gaussian white noise, B_Z is a standard Brownian motion, and the covariance is given explicitly. The proof proceeds through a stochastic Volterra representation of the intensity process (Proposition 4.1), followed by martingale approximations and a functional CLT for the cumulative intensity (Proposition 3.10). Functional CLTs are also proved for the associated shot noise processes (Theorem 3.13), and the results are applied to toxin accumulation in budding microbial populations in a host (Section 5).","tokens_in":34404,"tokens_out":16998,"duration_ms":155086,"significance":"If the main theorem is correct, this is a substantial contribution: it gives a genuinely measure-valued functional CLT for marked Hawkes processes with explicit covariance structure, separating the mark-wise white-noise fluctuations from the common Brownian fluctuation induced by self-excitation. It extends known CLTs for marked Hawkes processes (Karabash–Zhu) and provides new functional CLTs for Hawkes shot noise and for the integral of population in a microbial dynamics model. The paper contains detailed moment estimates and explicit constants, and the limiting objects are stated directly in terms of the model parameters with no fitted constants. The significance is tempered, however, by the fact that the central representation is imported from the authors' own unpublished preprint [39] and by an internal inconsistency in the displayed form of that representation.","major_comments":[{"comment":"The representation (4.6) as printed is not the representation used in the rest of the proof. Its second term is ∫_0^t R_H(t-s) µ0(t) ds, with the terminal time t inside the integrand, and the same terminal-time form appears in the first term of (4.3). Equation (4.7), which is obtained by integrating (4.6), uses instead the causal term ∫_0^{Tt} R_H(Tt-s) ds ∫_0^s µ0(r) dr, i.e. the integral of ∫_0^t R_H(t-s) µ0(s) ds. All subsequent estimates (Lemmas 4.2–4.4, Propositions 4.6–4.11) and the proof of Proposition 3.10 rely on the causal form. This is a load-bearing identity: the displayed equations must be corrected to the intended form, or, if the terminal-time form is genuinely what [39] supplies, the switch between the two representations must be justified. As it stands, the proof of the functional CLT is not self-consistent.","section":"Section 4, Prop. 4.1 and Eq. (4.6)"},{"comment":"Proposition 4.1 is imported from Theorem 2.2 of the authors' unpublished preprint [39], but the hypotheses of that theorem are neither stated nor checked against Condition 3.2. Because every martingale approximation in Section 4 starts from Eq. (4.6), the central proof is conditional on an unverified external result. The authors should prove Proposition 4.1 in this paper, or provide a complete verification that the conditions of [39, Theorem 2.2] follow from Condition 3.2, including the two-parameter kernel R and the relevant integrability conditions. Relying on an inaccessible self-cited result is not sufficient for a self-contained proof of the main theorem.","section":"Section 4, Prop. 4.1 / reference [39]"},{"comment":"The proof of Proposition 3.9 verifies convergence of the real-valued processes W^f_T for each f∈B(U) and identifies the covariance structure of the limit, but it does not explicitly establish tightness of the S(U)-valued processes needed for weak convergence in D([0,∞),S(U)). Since Theorem 3.11 is the central claim, the authors should supply the tightness argument for the measure-valued sequence, or give a precise reference showing that the one-dimensional projections established in the proof imply tightness in D([0,∞),S(U)).","section":"Section 4.1, proof of Proposition 3.9 / Theorem 3.11"}],"minor_comments":[{"comment":"The formula for |c_m|² contains the denominator |1−∑_{i=1}^d ‖φ(i)‖_{L1}|³, whereas the stability condition and the displayed |c_k|² both involve the weighted norm ∑_{i=1}^d ν_H({i})‖φ(i)‖_{L1}. As printed, the two denominators are inconsistent and the common-factor variance should be computed with the weighted norm.","section":"Example 3.16"},{"comment":"The independence argument at the end of the proof refers to 'N0(ds,du,dz) and N1(ds,du)'; the process N1 is never defined and should be N_I(ds,du).","section":"Section 4.1, proof of Proposition 3.9"},{"comment":"In the statement of inequality (4.29), the integration variable in the last term is written 'ν_H(dy)' while earlier in the same equation it is 'ν_H(du)'; this should be made uniform.","section":"Section 4.2, Lemma 4.8"},{"comment":"The symbol Φ T_i in the display following (5.9) appears to be a typo for Φ_i; the superscript T is otherwise unexplained.","section":"Section 5, Theorem 5.2"}],"recommendation":"major_revision","confidential_remarks":"The main issue is self-reliance on the authors' unpublished preprint [39] for the stochastic Volterra representation that carries the central proof. Even if Eq. (4.6) is a typo, the manuscript is not self-contained as submitted, and the internal inconsistency in the displayed representation must be resolved. This is fixable within the scope of the paper, so I do not recommend rejection, but the revision needs to be substantive rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you work on Hawkes processes. The paper delivers the first functional CLT for marked Hawkes point measures with immigration, and for the associated shot noise. The limit is a Gaussian white noise plus a lift of a correlated Brownian motion, with explicit covariance. That is a real advance over Karabash–Zhu (non-functional) and Bacry et al. (multivariate but unmarked).\n\nThe martingale approximation is handled with care. The estimates are explicit, the covariance formulas are given in terms of model parameters, and the application to budding microbes gives a functional version of Pakes's CLT. The LLN is derived as a corollary, which is tidy. The reliance on an unpublished preprint is visible and acknowledged, which is honest.\n\nNow the soft spot. Proposition 4.1, Eq. (4.6), as printed is wrong in a way that matters. The second term is ∫_0^t R_H(t-s) µ0(t) ds, where µ0(t) does not depend on s. That is not a causal convolution and makes no dimensional sense. The next display, (4.7), integrates to the causal form ∫_0^{Tt} R_H(Tt-s) ds ∫_0^s µ0(r) dr. So either (4.6) is a typo (µ0(s) instead of µ0(t)) or the proof silently switches to a different representation. All subsequent lemmas use the causal form. This is not a fatal mathematical error—the theorem is plausible—but it is exactly the kind of thing that must be fixed and verified.\n\nSecond, the representation is imported from Theorem 2.2 of the same authors' preprint [39]. The hypotheses of that theorem are never checked against Condition 3.2. The reader cannot tell whether the identity holds under the stated assumptions. That is a genuine gap in self-containedness, and the paper should address it before acceptance.\n\nMinor issue: some 'similar' estimates are delegated, e.g. in Lemmas 4.2–4.4, but those look like standard Burkholder-style arguments once the representation is fixed.\n\nWho should read it: anyone working on point process limit theorems, Hawkes processes, or stochastic Volterra representations. It deserves a serious referee. Send it to peer review; the recommendation should be major revision, primarily to correct the representation and either prove the needed version or cite a published/full proof.","headline":"A genuine new functional CLT for marked Hawkes measures, held back by a load-bearing typo in the Volterra representation and an unchecked dependency on an unpublished preprint.","tokens_in":34803,"tokens_out":3138,"would_cite":true,"duration_ms":31146,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G57","60F17","92B05","92D25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Marked Hawkes point measures fluctuate as Gaussian white noise plus a common Brownian factor.","keywords":["Hawkes point measure","marked Hawkes process","functional central limit theorem","functional law of large numbers","shot noise process","stochastic Volterra equation","Gaussian white noise","budding microbes in a host"],"falsifier":"Set $U=\\{1,2\\}$, take $\\varphi(t,u)=a_u e^{-b_u t}$ and constant immigration, and simulate the marked Hawkes process under Condition 3.2; the theorem predicts that the covariance of $\\sqrt{T}\\,\\bar N^T_{H,t}(\\{1\\})$ and $\\sqrt{T}\\,\\bar N^T_{H,t}(\\{2\\})$ converges to $\\sigma_Z^2\\,\\nu_H(\\{1\\})\\nu_H(\\{2\\})\\,t$, so a systematic discrepancy beyond Monte Carlo error would refute the claim.","tokens_in":33875,"feed_emoji":"🎲","tokens_out":9251,"duration_ms":91136,"temperature":0.7,"pith_summary":"The paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures: random counting measures in which each event carries a mark and the arrival rate depends on all past events and their marks. Under a stability condition, the rescaled point measure converges to a deterministic linear growth, and the fluctuation process converges to a Gaussian white noise plus a lifting of a one-dimensional Brownian motion. The Brownian term is a common factor across all mark sets and encodes the self-exciting feedback of event arrivals. If the result is correct, it gives explicit covariance formulas for the fluctuations of Hawkes-type counting processes and their shot noise, and it upgrades known central limit theorems to functional ones with applications to microbial population dynamics.","feed_headline":"Hawkes point measures converge to white noise plus Brownian motion","feed_subtitle":"A central limit theorem splits Hawkes fluctuations into mark-specific white noise and a common Brownian factor with explicit covariance.","key_machinery":"The central object is a stochastic Volterra representation of the intensity process: $Z(t)=\\mu_0(t)+\\int_0^t R_H(t-s)\\mu_0(s)\\,ds+\\int_0^t\\int_U R(t-s,u)\\,N_I(ds,du)+\\int_0^t\\int_U\\int_0^{Z(s-)} R(t-s,u)\\,\\tilde N_0(ds,du,dz)$, with $R=\\varphi+R_H*\\varphi$ the mean-impact kernel that resolves the branching structure. This representation rewrites the intensity as drift plus stochastic integrals against martingale measures; the martingale integrals are approximated by $\\|R(u)\\|_{L^1}$-weighted martingales, whose weak convergence yields both the Gaussian white noise and the Brownian component. The function $R(t,u)$ therefore carries the decomposition of fluctuations into mark-specific white noise and a common Brownian factor.","core_discovery":"The central claim is that, under Condition 3.2, $\\sqrt{T}\\,\\bar N^T_{H,t}(A)$ converges weakly in $D([0,\\infty),S(U))$ to $W_{H,t}(A)+\\sigma_Z B_Z(t)\\nu_H(A)$, where $W_H$ is a Gaussian white noise on $U$ with intensity $\\lambda_I\\|R_I\\|_{L^1}\\,dt\\,\\nu_H(du)$ and $B_Z$ is a standard Brownian motion built from both the Hawkes white noise and an independent immigration white noise. The variance is $\\sigma_Z^2=\\lambda_I\\big(\\|R_I\\|_{L^1}\\nu_H(\\|\\varphi(\\cdot)\\|_{L^1}^2)+\\nu_I(\\|\\varphi(\\cdot)\\|_{L^1}^2)\\big)\\big/|1-\\|\\varphi_H\\|_{L^1}|^2$. The same decomposition drives functional central limit theorems for shot noise processes and, in the budding-microbe application, for the joint process of cumulative budding rate and total toxin release.","pith_inferences":["The paper leaves implicit that the covariance between disjoint mark sets is purely Brownian, which gives a testable factor structure for Hawkes-type data.","The same Volterra-martingale route would likely produce an analogous decomposition for other self-exciting point processes whenever a resolvent kernel exists, even outside the Hawkes-specific branching structure.","The long-memory remark at the end of the paper points to a concrete open test: replace the light-tail kernel condition by regular variation and check whether the Brownian factor becomes a Gaussian process with Riemann-Liouville-type covariance.","A direct practical use is to turn the explicit variance formula into finite-sample confidence bands for estimated Hawkes intensities; the formula predicts the exact $1/\\sqrt{T}$ scaling of estimation error under self-excitation."],"forward_implications":["The explicit limit gives covariance formulas: for disjoint mark sets $A_1,A_2$, the asymptotic covariance of the two scaled count fluctuations is $\\sigma_Z^2\\,t\\,\\nu_H(A_1)\\nu_H(A_2)$, so correlatedness across marks is governed by the Brownian factor.","The functional CLT for shot noise processes does not rely on the law of large numbers; the LLN follows as a corollary, making the fluctuation result the primary statement.","For standard marked Hawkes processes with constant immigration, the result yields a functional CLT whose one-dimensional version recovers a known CLT for marked Hawkes processes.","In the budding-microbe model, the theorem provides a joint functional CLT for cumulative budding rate and total toxin, and, when toxins are released at unit rate, a functional CLT for the integral of population, extending a previously known CLT to a process-level statement."],"supporting_citations":[{"why":"Supplies the stochastic Volterra representation of the intensity process, on which Proposition 4.1 and all subsequent martingale approximations rest.","marker":"[39]"},{"why":"Introduces Hawkes random measures, the framework used to link the marked Hawkes process to a two-parameter intensity and to derive the Volterra representation.","marker":"[19]"},{"why":"Provides the Lindeberg-Feller criteria and tightness results used to prove weak convergence of the approximating martingales.","marker":"[21]"},{"why":"Supplies the martingale-measure machinery and the characterization of Gaussian white noise used to identify the limit.","marker":"[37]"},{"why":"Gives the Volterra resolvent theory used to solve the expectation equation for the intensity and to define the kernel R.","marker":"[15]"},{"why":"Proves the one-dimensional CLT for marked Hawkes processes that the present functional CLT extends.","marker":"[25]"},{"why":"Provides the functional CLT for multivariate Hawkes processes that serves as the benchmark for the measure-valued generalization.","marker":"[2]"},{"why":"Proves the CLT for the integral of a branching population, which the microbe application upgrades to a functional CLT.","marker":"[31]"}],"fun_headline_variants":["Marked Hawkes point measures split into white noise plus Brownian motion","Functional CLT for Hawkes point measures: white noise + Brownian motion","Hawkes point measures: white noise and Brownian decomposition","Budding microbes obey Hawkes limit theorems with Brownian noise","White noise plus Brownian: limit law for marked Hawkes measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the stochastic Volterra representation of the intensity process taken from an earlier preprint; if the hypotheses of that theorem fail for a given kernel, the proof of the functional limit theorems no longer goes through.","fun_headline_variants_meta":{"raw":{"variants":["Marked Hawkes point measures split into white noise plus Brownian motion","Functional CLT for Hawkes point measures: white noise + Brownian motion","Hawkes point measures: white noise and Brownian decomposition","Budding microbes obey Hawkes limit theorems with Brownian noise","White noise plus Brownian: limit law for marked Hawkes measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00087,"raw_usage":{"total_tokens":3713,"prompt_tokens":835,"completion_tokens":2878,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":451,"completion_tokens_details":{"reasoning_tokens":2787}},"tokens_in":451,"tokens_out":2878,"duration_ms":19208,"temperature":1.0,"reasoning_tokens":2787,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:35:56.072162+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $U=\\{1,2\\}$, take $\\varphi(t,u)=a_u e^{-b_u t}$ and constant immigration, and simulate the marked Hawkes process under Condition 3.2; the theorem predicts that the covariance of $\\sqrt{T}\\,\\bar N^T_{H,t}(\\{1\\})$ and $\\sqrt{T}\\,\\bar N^T_{H,t}(\\{2\\})$ converges to $\\sigma_Z^2\\,\\nu_H(\\{1\\})\\nu_H(\\{2\\})\\,t$, so a systematic discrepancy beyond Monte Carlo error would refute the claim.","supporting_citations":[{"cited_title":"Scaling Limits for Crump-Mode-Jagers Processes with Immigration via Stochastic Volterra Equations","cited_arxiv_id":"1809.05931","evidence_quote":"Supplies the stochastic Volterra representation of the intensity process, on which Proposition 4.1 and all subsequent martingale approximations rest."},{"cited_title":"Horst and W","cited_arxiv_id":null,"evidence_quote":"Introduces Hawkes random measures, the framework used to link the marked Hawkes process to a two-parameter intensity and to derive the Volterra representation."},{"cited_title":"Jacod and A","cited_arxiv_id":null,"evidence_quote":"Provides the Lindeberg-Feller criteria and tightness results used to prove weak convergence of the approximating martingales."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the martingale-measure machinery and the characterization of Gaussian white noise used to identify the limit."},{"cited_title":"Gripenberg, S.-O","cited_arxiv_id":null,"evidence_quote":"Gives the Volterra resolvent theory used to solve the expectation equation for the intensity and to define the kernel R."},{"cited_title":"Karabash and L","cited_arxiv_id":null,"evidence_quote":"Proves the one-dimensional CLT for marked Hawkes processes that the present functional CLT extends."},{"cited_title":"Bacry, S","cited_arxiv_id":null,"evidence_quote":"Provides the functional CLT for multivariate Hawkes processes that serves as the benchmark for the measure-valued generalization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the CLT for the integral of a branching population, which the microbe application upgrades to a functional CLT."}],"review_version":1}