{"id":"d64c48e0-3492-45c0-9b64-e964611f8885","arxiv_id":"2412.14459","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Asymptotically critical multivariate Hawkes processes converge to the unique weak solution of a stochastic Volterra equation with a measure kernel, characterized by an admissible pair (K, Φ).","lead":"This paper proves that nearly critical multivariate Hawkes processes, after a suitable time and space rescaling, converge weakly to a multidimensional stochastic Volterra equation whose kernel is a potential measure of a matrix-valued extended Bernstein function. If the proofs hold, the result gives a complete one-to-one classification of all non-degenerate Hawkes scaling limits, unifying the known light-tailed and heavy-tailed cases.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The one-to-one correspondence is only proved under Assumption 2.8; non-diagonal Π(0) leaves uniqueness of (2.21) unproved, as Remark 2.21 concedes.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing point: Assumption 2.8 (diagonal Π(0)) is used in Lemma 4.4 to prove uniqueness of the Riccati-Volterra equation (2.21), and Lemma 5.6 converts that uniqueness into uniqueness in law of accumulation points in Theorem 2.9. My independent reading of the proof confirms this: the only place where the atom of Π at zero is controlled is in (4.15)-(4.18), and there the argument requires Π_ji(0)=0 for i≠j. Remark 2.21 explicitly concedes that the uniqueness proof 'will no longer work' without the diagonal assumption, leaving only subsequential convergence. Since Section 1.3 states the one-to-one correspondence without this qualification, the paper's advertised central claim is not fully supported by its proofs. I found an explicit admissible couple with non-diagonal Π(0), showing the assumption is not vacuous; whether uniqueness actually fails for that example remains open, which is why the appropriate action is a concrete test rather than outright rejection. The paper's conditional theorems under Assumption 2.8 appear internally coherent, and the author's own remark demonstrates awareness of the limitation, so the reader's CONDITIONAL verdict is appropriate and no change is needed.","tokens_in":46642,"tokens_out":12816,"duration_ms":113254,"concrete_test":"Verify whether Lemma 4.4's uniqueness claim remains true for the explicit non-diagonal atom example K=[[0,1],[1,0]], Φ(λ)=(1-2e^{-λ})I_2, for which Π(0)=[[1/2,1/2],[1/2,1/2]]. Compute the t=0 algebraic system V_i=f_i(0)+1/4Σ_j(V_j+h_j(0))^2 and classify all solutions with Re(V_i)≤0 for all admissible f(0),h(0); then attempt an independent proof of Lemma 4.4 without Assumption 2.8 on this example. If two distinct nonpositive local solutions exist, the unqualified one-to-one correspondence fails; if uniqueness still holds, the diagonal assumption is a proof artifact that the paper must repair and state explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in Section 1.3 asserts an unqualified one-to-one correspondence between admissible couples (K,Φ) and unique scaling limits. The proof of uniqueness of the limit law runs through Lemma 4.4, which proves uniqueness of the Riccati-Volterra equation (2.21) only by using Assumption 2.8 that Π(0) is diagonal: in (4.15)-(4.18) the atom Π(0) is separated only when off-diagonal entries vanish. Remark 2.21(1) explicitly states that if Π(0) is non-diagonal the proof 'will no longer work' and only subsequential convergence along accumulation points is available. Consequently Theorem 2.9 and the advertised converse in Section 1.3 are not established for all admissible couples; they are conditional on Assumption 2.8, or on the extra hypotheses of Remark 2.21(2). The assumption is non-vacuous: for K=[[0,1],[1,0]] and Φ(λ)=(1-2e^{-λ})I_2, the real Schur decomposition with Q=2^{-1/2}[[1,1],[1,-1]] gives LΠ(λ)=(1-2e^{-λ})^{-1}Qdiag(1,0)Q^T, so Π(0)=[[1/2,1/2],[1/2,1/2]], a non-diagonal atom. Thus the gap between the advertised conclusion and the proven theorem is real, though it does not invalidate the results that explicitly assume diagonal Π(0).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops full scaling limit theorems for asymptotically critical multivariate Hawkes processes. Under a Condition 2.3 specifying the asymptotic criticality through a limit matrix K and a matrix-valued extended Bernstein function Φ, the author proves joint weak convergence of the rescaled integrated intensity, point process, and compensated point process in the S-topology (Theorem 2.9). The limit process (Ξ,M) is then characterized in three ways: by an exponential-affine Fourier-Laplace representation in terms of the unique solution of a Riccati-Volterra equation with measure kernel (Theorem 2.12), by the stochastic Volterra equation Ξ = Υ + Π ∗ M (Theorem 2.13), and by an alternate representation separating the drift parameter bΦ (Theorem 2.19). The paper also gives regularity results and a concrete example. The advertised central conclusion in Section 1.3 is a one-to-one correspondence between admissible couples (K,Φ) and unique scaling limits solving the corresponding Volterra equation.","tokens_in":46898,"tokens_out":16712,"duration_ms":134301,"significance":"If the results hold in their advertised generality, this is a substantial contribution: it provides multivariate scaling limits for Hawkes processes beyond the Markovian or light-tailed settings, introduces potential measures of matrix-valued extended Bernstein functions as convolution kernels, and establishes well-posedness of Riccati-Volterra equations with measure kernels, which is genuinely nonstandard. The paper's strengths include detailed proofs of the main theorems, explicit assumptions with no fitted parameters, a constructive existence result for the approximating Hawkes sequences, and a useful potential-theoretic perspective connecting the limit to Lévy subordinators. However, the most important caveat is that the uniqueness of the limit law, and therefore the advertised one-to-one correspondence, is proved only under Assumption 2.8 that Π(0) is diagonal. The paper itself acknowledges in Remark 2.21 that without this assumption only subsequential convergence is obtained. This makes the unconditional claims in the abstract and Section 1.3 stronger than the proved theorems, and the necessary repair is not merely cosmetic.","major_comments":[{"comment":"The advertised one-to-one correspondence is only established under Assumption 2.8. The uniqueness of accumulation points in Theorem 2.9 is obtained by combining Lemma 4.4, which proves uniqueness for the Riccati-Volterra equation (2.21), with Lemma 5.6, which identifies all accumulation points through the common Fourier-Laplace functional (2.22). Lemma 4.4 uses the diagonality of Π(0) in equations (4.15)-(4.18) to isolate the atom of the measure; for non-diagonal Π(0) the off-diagonal atom terms do not separate in this way. Remark 2.21(1) concedes that in this case the proof of uniqueness 'will no longer work' and only subsequential convergence along accumulation points is available. The gap is non-vacuous: for K = [[0,1],[1,0]] and Φ(λ) = (1-2e^{-λ})I_2, the real Schur decomposition with Q = 2^{-1/2}[[1,1],[1,-1]] gives LΠ(λ) = (1-2e^{-λ})^{-1} Q diag(1,0) Q^T, so Π(0) = [[1/2,1/2],[1/2,1/2]], a non-diagonal atom. Thus the abstract and Section 1.3 statements that every admissible couple yields a unique scaling limit go beyond what is proved; the results are conditional on Assumption 2.8 or on the additional hypotheses of Remark 2.21(2). I recommend either proving uniqueness without the diagonal assumption or restating the main theorems and the concluding correspondence so that the diagonal condition is explicitly included.","section":"Assumption 2.8; Lemma 4.4; Remark 2.21"}],"minor_comments":[{"comment":"The symbol Π is used both for the potential measure and for its distribution function Π(t) = Π([0,t]). In (5.15) the term Π ∗ M(T) is the Lebesgue convolution with the function Π(·), while the Fubini identity immediately after it interprets Π ∗ M(t) as the measure convolution. These two objects are equal, but only after the displayed identity is verified with the distribution-function notation made explicit. As written, the sentence 'By Fubini's theorem, Π ∗ M(T) = ∫_0^T Π ∗ M(t)dt' is ambiguous and potentially confusing; please clarify the two meanings of Π ∗ M.","section":"Lemma 4.4, last paragraph"},{"comment":"In the reduction to the case h ≠ 0, the equation written for V^h should be V^h = h + f ∗ Π + 1/2 (V^h)^2 ∗ Π, and similarly for V^{*,h}; the printed equation V^h = f + h + 1/2 (V^h)^2 ∗ Π drops the convolution on f. The uniqueness argument is unaffected because the f-term cancels in the difference, but the displayed equation is not the equation satisfied by V + h.","section":"Corollaries 2.17 and 2.20; Remark 2.24"},{"comment":"Several auxiliary results are stated without full proofs. Corollary 2.17 says 'The detailed proof of this corollary is omitted', Corollary 2.20 says 'The proof is omitted', and Remark 2.24 announces an intensity convergence result by 'generalizing the proof' of [37] or using [68]. These statements may be acceptable if the arguments are indeed routine, but for a journal submission it would be helpful to provide sketches or precise references that indicate exactly which arguments in [37] and [68] are being extended.","section":"Notation and Proposition C.3"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the mismatch between the unconditional one-to-one correspondence advertised in the abstract and Section 1.3 and the conditional proof under Assumption 2.8. The author's own Remark 2.21 is the strongest evidence of the gap. I would ask for a revision that either proves uniqueness for non-diagonal Π(0) or explicitly restricts the main theorems and the concluding correspondence to the diagonal case. The rest of the paper appears technically substantial, and the conditional results are likely publishable if the claims are aligned with the proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about Xu's paper: it is a real advance in the scaling-limit theory of multivariate Hawkes processes, and it is not quite the complete classification its introduction advertises. The advertised one-to-one correspondence between admissible couples (K,Φ) and unique limits is proved only under Assumption 2.8, that Π(0) is diagonal. The paper's own Remark 2.21 says the uniqueness proof fails without it and only subsequential convergence holds. The stress-test gives a concrete non-vacuous example (K=[[0,1],[1,0]], Φ(λ)=(1-2e^{-λ})I_2) with non-diagonal Π(0), so the gap is real. To be fair, the paper is explicit about this limitation; the overstatement is in the abstract and Section 1.3, not in the proofs.\n\nWhat is genuinely new: the multivariate measure-kernel framework that handles light- and heavy-tailed kernels in one setting; the construction of the (K,Φ)-potential measure via real Schur decomposition; the well-posedness of Riccati-Volterra equations with measure kernel; the exponential-affine representation of the Fourier-Laplace functional; and the alternate SVE representations separating the drift parameter. The proofs are detailed and I found no circularity or fitted parameters. The reliance on earlier work by the same author for the martingale representation is legitimate and properly cited.\n\nThe soft spots beyond Assumption 2.8 are minor but real: Corollaries 2.17 and 2.20 are stated without proof, and Remark 2.24's convergence claim is asserted rather than demonstrated. The paper would benefit from stating the main theorem with the diagonal condition in the theorem statement and moving the unconditional converse to a 'conditional' form.\n\nWho is this for: researchers in point-process scaling limits, Volterra equations, and branching processes. It deserves a serious referee; the central machinery is substantial and mostly correct under the stated assumption. I would send it to review, but the referee should push for a clearer statement about Assumption 2.8 and for proofs of the omitted auxiliaries.","headline":"Substantial and mostly new multivariate Hawkes scaling-limit machinery, but the advertised one-to-one correspondence is proven only under the diagonal-atom assumption.","tokens_in":47476,"tokens_out":2529,"would_cite":true,"duration_ms":20913,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G55","60F17","60H20","45D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that asymptotically critical multivariate Hawkes processes converge, after rescaling, to the unique solution of a stochastic Volterra equation with a measure kernel, and that every admissible limiting pair arises this way.","keywords":["Hawkes process","scaling limit theorem","stochastic Volterra equation","measure kernel","extended Bernstein function","potential measure","Riccati-Volterra equation","branching process with immigration"],"falsifier":"Construct a two-type admissible couple (K,Φ) with Π({0}) non-diagonal and an asymptotically critical sequence satisfying Condition 2.3; if two different accumulation points have different laws at some fixed time, the claimed sequence-level weak convergence fails. A shorter check is to exhibit two distinct global solutions of the Riccati-Volterra equation V=W∗Π for such Π, which would directly contradict the uniqueness theorem.","tokens_in":46368,"feed_emoji":"📈","tokens_out":7434,"duration_ms":64268,"temperature":0.7,"pith_summary":"This paper establishes full scaling limit theorems for multivariate Hawkes processes that are asymptotically critical: after rescaling time by n and space by nθ_n, the point process and its compensated martingale converge weakly to a limit pair (Ξ,M) governed by a multidimensional stochastic Volterra equation with a measure kernel. The kernel is the potential measure Π(dt) associated to a matrix-valued extended Bernstein function Φ, so the limit is fixed by an admissible couple (K,Φ) consisting of a non-negative matrix with unit spectral radius and such a Bernstein function. The paper proves the correspondence is one-to-one: every admissible couple yields a unique limit, and every non-degenerate scaling limit arises from some admissible couple. This matters because it unifies previously separate light-tailed and heavy-tailed limit regimes and identifies the limits as a class of non-Markovian branching processes with immigration whose Laplace functionals are computable through a Riccati-Volterra equation.","feed_headline":"All scaling limits of Hawkes processes are Volterra equations","feed_subtitle":"Every admissible kernel pair (K,Φ) maps to exactly one limit process, unifying light- and heavy-tailed cases.","key_machinery":"The carrying object is the (K,Φ)-potential measure Π(dt), a matrix-valued measure on R+ whose Laplace transform is built from the Schur decomposition of K and the inverse of the block ϕ(λ) determined by Φ; it is the vague limit of the rescaled resolvent densities R(n)(t)dt. The proof mechanism is the convergence of exponential-affine representations: the rescaled Fourier-Laplace functional of each Hawkes process is written exactly in terms of a nonlinear Volterra equation, and these equations are shown to converge to the Riccati-Volterra equation V=W∗Π, W=f+½(V+h)², with the measure Π(dt) acting as the convolution kernel. Uniqueness of that Riccati-Volterra equation is the step that turns relative compactness of all accumulation points into weak convergence of the entire sequence.","core_discovery":"The paper's central claim is that the rescaled Hawkes processes converge, in law and after a time-space rescaling, to the unique weak solution (Ξ,M) of Ξ(t)=Υ(t)+Π∗M(t), where Υ records the accumulated exogenous input, M is the limiting martingale, and Π(dt) is the (K,Φ)-potential measure. Uniqueness is obtained through an exponential-affine representation of the Fourier-Laplace functional: E[exp{f∗dΞ(T)+h∗dM(T)}]=exp{W∗dΥ(T)}, where V∈D(R+;C−) is the unique global solution to the Riccati-Volterra equation V=W∗Π with W=f+½(V+h)². Conversely, the paper constructs, for any admissible (K,Φ), an asymptotically critical sequence of Hawkes processes whose scaling limit is exactly that pair, so the admissible couples classify all non-degenerate limits. The same machinery yields an alternate Volterra representation that separates the drift parameter bΦ from the volatility and jump parameters, and it gives derivative-process regularity when the potential density is locally square-integrable.","pith_inferences":["The paper leaves the non-diagonal-atom case to subsequential convergence; that case is the natural stress test, since any counterexample with two different accumulation points would mark the precise boundary of the weak convergence theorem.","The classification suggests a reverse-engineering recipe: choose a desired Volterra limit, read off (K,Φ), and build Hawkes kernels that realize it; the example with Φ(λ)=b+c(λ+β)^α already recovers fractional-Heston-type equations.","One might expect the Riccati-Volterra uniqueness proof to extend beyond Assumption 2.8 by a more general contraction involving the full matrix Π(0), which would remove the stated bottleneck and upgrade many subsequential results to full convergence.","A possible extension to marked or nonlinear Hawkes processes would carry the same measure-kernel Riccati equation over, potentially producing jump-type rough volatility limits."],"forward_implications":["Every non-degenerate scaling limit of a multivariate Hawkes process is characterized uniquely by an admissible couple (K,Φ), so studying such limits reduces to choosing the potential measure.","The limit pair (Ξ,M) inherits the branching property and behaves as a generally non-Markovian continuous-state branching process with immigration, with Laplace functional computed from the Riccati-Volterra solution.","When either Ξ or M is continuous, the convergence holds in the Skorokhod J1 topology and M is a time-changed Brownian motion, M=B∘Ξ.","If the potential measure has a locally square-integrable density and Υ is differentiable, the limit intensity has a predictable derivative ξ solving a stochastic Volterra equation with √ξ diffusion coefficient.","The equivalent representation separating bΦ gives a criticality criterion for Ξ: subcritical when Π(∞)<∞, critical when λΠ=0 and Π(∞)=∞, and supercritical when λΠ>0."],"supporting_citations":[{"why":"supplies the univariate nearly unstable Hawkes scaling limit result that this paper extends to multivariate processes.","marker":"[41]"},{"why":"supplies the heavy-tailed univariate scaling limit whose rough-fractional analogue the multivariate theorem recovers.","marker":"[42]"},{"why":"provides the affine Volterra process framework and duality method used for uniqueness in law of the limit pair.","marker":"[3]"},{"why":"provides the functional limit theorems and the martingale representation lemma for the Hawkes intensity used throughout.","marker":"[36]"},{"why":"provides weak convergence results for stochastic integrals and the S-topology needed for tightness and weak convergence.","marker":"[43]"},{"why":"gives the Bernstein function representation and potential measure theory that identify the kernel Π(dt).","marker":"[65]"},{"why":"supplies the Volterra equations with measure kernels and resolvent results used to establish well-posedness.","marker":"[29]"}],"fun_headline_variants":["Hawkes scaling limits are all Volterra equations","Every Hawkes limit solves a Volterra equation","Measure-kernel Volterra equations classify Hawkes limits","Riccati-Volterra solves the Hawkes scaling limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the potential measure Π has no off-diagonal atom at zero, i.e., Π(0) is diagonal; without it the paper's proof of uniqueness of the Riccati-Volterra equation no longer works, and only subsequential convergence is obtained.","fun_headline_variants_meta":{"raw":{"variants":["Hawkes scaling limits are all Volterra equations","Every Hawkes limit solves a Volterra equation","Measure-kernel Volterra equations classify Hawkes limits","Riccati-Volterra solves the Hawkes scaling limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0006,"raw_usage":{"total_tokens":2790,"prompt_tokens":917,"completion_tokens":1873,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":1810}},"tokens_in":533,"tokens_out":1873,"duration_ms":11516,"temperature":1.0,"reasoning_tokens":1810,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:13:21.853574+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a two-type admissible couple (K,Φ) with Π({0}) non-diagonal and an asymptotically critical sequence satisfying Condition 2.3; if two different accumulation points have different laws at some fixed time, the claimed sequence-level weak convergence fails. A shorter check is to exhibit two distinct global solutions of the Riccati-Volterra equation V=W∗Π for such Π, which would directly contradict the uniqueness theorem.","supporting_citations":[{"cited_title":"Abi Jaber, M","cited_arxiv_id":null,"evidence_quote":"provides the affine Volterra process framework and duality method used for uniqueness in law of the limit pair."},{"cited_title":"Horst and W","cited_arxiv_id":null,"evidence_quote":"provides the functional limit theorems and the martingale representation lemma for the Hawkes intensity used throughout."},{"cited_title":"Jakubowski","cited_arxiv_id":null,"evidence_quote":"provides weak convergence results for stochastic integrals and the S-topology needed for tightness and weak convergence."},{"cited_title":"Gripenberg, S","cited_arxiv_id":null,"evidence_quote":"supplies the Volterra equations with measure kernels and resolvent results used to establish well-posedness."}],"review_version":1}