{"id":"3a93b0a1-20cb-42ab-91b8-47af703f1e09","arxiv_id":"2501.11648","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nearly unstable Hawkes processes rescale to affine stochastic Volterra diffusions, and mean-field Hawkes systems exhibit synchronization, conditional independence, or extinction depending on n(1-||phi^n||)^2.","lead":"This paper proves that nearly unstable Hawkes processes, in which the self-exciting kernel is close to critical, converge under rescaling to stochastic Volterra diffusions with affine structure, covering a much broader class of kernels than earlier results. It also derives propagation of chaos for systems of many interacting Hawkes particles, with three distinct macroscopic regimes depending on how fast criticality is approached.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's regime parameter n(1-||phi^n||)^2 is not tied to the theorem's zeta=lim n beta_n^2; Condition 2.1 permits beta_n much larger than 1-||phi^n||, so a valid nearly unstable family can have n(1-||phi||)^2 -> 0 while n beta_n^2 -> infinity.","rationale":"The reader's weakest assumption was Condition 2.1: the L1 criticality condition does not imply the scaling compactness and L2 bound. My concern is sharper: even when Condition 2.1 holds, the scaling sequence beta_n is a free parameter, and the abstract's replacement of zeta=lim n beta_n^2 by n(1-||phi^n||_{L1})^2 can be false. The explicit stable-density family satisfies all theorem assumptions but lies in the extinction regime according to the theorem while being classified as the synchronized regime by the abstract. This does not invalidate Theorem 2.2 or the conditional statements of Theorems 3.1-3.2, whose proofs appear coherent; it means the abstract and Remark 3.3 need correction, and the paper should either restrict to cases where beta_n is comparable to 1-||phi^n|| or state the regimes only in terms of the beta_n appearing in Condition 2.1. The verdict remains conditional, matching the reader's assessment, but the required revision is more substantial than a wording change.","tokens_in":28657,"tokens_out":41335,"duration_ms":443453,"concrete_test":"Verify the explicit counterexample for alpha=3/4: take phi the positive stable density with Laplace transform e^{-z^{3/4}}, set a_n=1-n^{-2/3}, b_n=n^{4/9}, beta_n=n^{-1/3}, and mu_0^n=n^{1/3}. Compute beta_n||psi_n||_{L^2_T} and beta_n psi_n(t) dt on [0,T]; for large n the latter should be close to (1/Gamma(3/4)) t^{-1/4} dt, whose L2 norm is finite. Then check n beta_n^2=n^{1/3}->infinity while n(1-a_n)^2=n^{-1/3}->0. If both hold, the abstract's regime statement is false for a sequence satisfying all hypotheses of Theorem 3.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 states the trichotomy in terms of zeta = lim n beta_n^2, where beta_n is the scaling sequence appearing in Condition 2.1. The abstract and Remark 3.3 instead phrase the three regimes in terms of n(1-||phi^n||_{L1})^2. These two parameters are not interchangeable: Condition 2.1 never forces beta_n to be comparable to 1-||phi^n||_{L1}. When the limiting measure F has infinite total mass, equivalently the Bernstein exponent Phi satisfies Phi(0)=0, Lemma 2.4 gives Lphi_n(z)=1-beta_n/LF(z)+o(beta_n), so 1-||phi^n||_{L1}=o(beta_n). A concrete family is phi_n(t)=a_n b_n phi(b_n t), where phi is the one-sided alpha-stable density with Lphi(z)=e^{-c z^alpha}, alpha in (1/2,1), with a_n=1-n^{-2/3}, b_n=n^{1/(3 alpha)}, beta_n=b_n^{-alpha}=n^{-1/3}. Then Condition 2.1 holds with limit LF(z)=1/(c z^alpha), and the L2 bound beta_n||psi_n||_{L2_T} is finite because on [0,T] the rescaled resolvent behaves like t^{alpha-1}, which is square-integrable for alpha>1/2. Taking mu_0^n = a n^{1/3} gives beta_n mu_0^n -> a>0, so Theorem 3.1 applies. But n beta_n^2 = n^{1/3} -> infinity, the extinction regime, while n(1-||phi_n||_1)^2 = n^{-1/3} -> 0, which the abstract labels as the synchronized regime. Thus the advertised phase diagram is not determined by the L1 gap; it depends on a scaling choice whose relation to 1-||phi^n|| is left unspecified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies scaling limits of nearly unstable Hawkes processes. Under Condition 2.1, which postulates the existence of a vanishing sequence β_n such that the rescaled resolvent measures β_n ψ^n_{ij}(t)dt converge vaguely to a σ-finite limit F_{ij} with a uniform L2 bound, Theorem 2.2 shows that the rescaled compensator, counting process, and martingale are C-tight and every limit point satisfies the affine stochastic Volterra equation X = F([0,t])·a + f*Z with Z_i = B_i∘X_i. The paper then considers a mean-field system of n Hawkes particles with a common intensity and, under the same Condition 2.1, establishes propagation of chaos (Theorems 3.1 and 3.2) with three regimes governed by ζ = lim n β_n^2: complete synchronization for ζ=0, conditional independence with Poisson-type idiosyncratic components for ζ∈(0,∞), and extinction for ζ=∞. The univariate case is further characterized in terms of Bernstein functions.","tokens_in":29108,"tokens_out":20372,"duration_ms":191996,"significance":"The main convergence argument is rigorous and economical: the proof of Theorem 2.2 combines moment estimates, Kolmogorov–Chentsov tightness, and the martingale representation theorem, and it does not require the restrictive kernel form φ^n = a_n φ used in earlier work. The mean-field trichotomy is a natural and nontrivial extension of Delattre–Fournier–Hoffmann, and the empirical measure convergence in Theorem 3.2 is a useful strengthening. The paper also gives a clean Bernstein-function parametrization of the possible limits in the univariate case. However, the advertised regime parameter in the abstract is not the one appearing in the theorems, and the abstract's 'mild' condition is weaker than Condition 2.1; these mismatches affect how the results can be applied and need to be fixed.","major_comments":[{"comment":"The abstract states that the three mean-field regimes depend on the asymptotics of n(1-||φ^n||_{L1})^2, but Theorem 3.1 and Remark 3.3 state that the trichotomy is governed by ζ = lim n β_n^2, where β_n is the scaling sequence appearing in Condition 2.1. Condition 2.1 does not imply that β_n is comparable to 1-||φ^n||_{L1}. In fact, by Lemma 2.4(3), 1-||φ^n||_{L1} = β_n/LF(0) + o(β_n); when LF(0)=∞ this quantity is o(β_n). A concrete such family is φ_n(t) = a_n b_n φ(b_n t), where φ is the one-sided α-stable density with Laplace transform e^{-c z^α}, α∈(1/2,1), a_n=1-n^{-2/3}, b_n=n^{1/(3α)}, and β_n=n^{-1/3}. For this family Condition 2.1 holds (the rescaled resolvent converges to the density with Laplace transform 1/(c z^α) and lies in L^2 on [0,T]), and with μ_0^n = a n^{1/3} we have β_n μ_0^n → a. But nβ_n^2 = n^{1/3} → ∞, so Theorem 3.1 gives extinction, while n(1-||φ^n||_{L1})^2 = n^{-1/3} → 0, which the abstract labels as the synchronized regime. Thus the abstract's phase diagram is not a consequence of the theorems and is in fact false for this family.","section":"Abstract and Theorem 3.1"},{"comment":"Condition 2.1 requires more than ||φ^n||_{L1} → 1: it requires the existence of β_n → 0 with sup_n β_n ||ψ^n_{ij}||_{L^2_T} < ∞ and vague convergence of β_n ψ^n_{ij}(t)dt. The abstract's phrase 'mild asymptotic criticality condition, specifically ||φ^n||_{L1} → 1' is therefore misleading. The paper itself notes in Section 2.2 that a necessary and sufficient condition for the L^2 bound is unknown and only provides a sufficient condition in dimension one (boundedness of ||φ^n||_{L^2_T}). Since Theorem 2.2 is the basis for Theorems 3.1 and 3.2, the abstract overstates the domain of applicability.","section":"Abstract and Section 2.1"},{"comment":"Remark 3.3 contains an incorrect assertion in the ζ=∞ case. The remark defines U_i = U for all i, where U is the aggregate compensator limit from Theorem 2.2, and then concludes 'Xi = Zi = Ui = 0'. But Theorem 2.2, applied to the aggregate Hawkes process, gives U = X with X(t) = F([0,t])a + f*Z(t); when β_n μ_0^n → a>0 and F is non-degenerate, X(t) ≥ F([0,t])a > 0 for t>0. The individual compensator Λ^{(n)}_i equals Λ^{(n)} (the aggregate rescaled compensator), so its limit is U = X, not 0. The theorem's actual conclusion in case (3) concerns only X_i and Z_i, which can vanish even though U is non-zero, because the martingale term in (3.13) is not tight after multiplication by the diverging factor √(nβ_n^2). The remark's heuristic should be corrected to avoid claiming Ui=0.","section":"Remark 3.3"}],"minor_comments":[{"comment":"The statement and proof of Proposition 2.5 write ρ(||g||_{L^2}) in the conclusion, but the argument uses ||g||_{L^1}; the L^2 norm is not defined for general g∈L^1. Please replace L^2 by L^1 in the statement and in the last displayed inequality.","section":"Section 2.2, Proposition 2.5"},{"comment":"The third case of Theorem 3.1 is labeled '(2)' instead of '(3)'; the numbering should be corrected.","section":"Theorem 3.1"},{"comment":"The proof uses the process '~λ^n(t)' without defining it; this should be defined as the intensity of the auxiliary process for particles K+1,...,n, or the proof should be rewritten using the previously defined θ^n.","section":"Section 5.2, proof of Proposition 5.3"},{"comment":"The notation ||F||_{L^1_t} is nonstandard for the cumulative measure F([0,t]) of the σ-finite measure F; consider writing F([0,t]) or F(t) throughout the paper to avoid confusion with a function norm.","section":"Section 2.1, Eq. (2.3)"},{"comment":"The phrase 'abstracted-valued random variables' should be 'abstract-valued random variables'.","section":"Section 1.2"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the paper appears sound: the proofs of Theorems 2.2, 3.1, and 3.2 are built on standard and largely complete arguments. The main problems are in the presentation of the results: the abstract's regime parameter is not the one in the theorems, the abstract's 'mild' condition is not sufficient for Condition 2.1, and Remark 3.3 contains a false statement about Ui in the extinction case. These are fixable in a revision, but they currently misrepresent the paper's contributions. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2501.11648. The mathematical core is worth taking seriously: Theorem 2.2 is a clean conditional scaling limit for nearly unstable Hawkes processes, and the Bernstein-function characterization in Lemma 2.4 genuinely unifies the Jaisson-Rosenbaum exponential and rough cases. The mean-field trichotomy in Theorems 3.1-3.2 is new relative to Delattre-Fournier-Hoffmann, and the coupling argument in Section 5 is well constructed. If you work on Hawkes asymptotics, this is a paper you need to know.\n\nBut the paper as written oversells itself in a way that matters. The abstract says the results hold under a 'mild asymptotic criticality condition, specifically ||phi^n||_L1 -> 1', and that the three mean-field regimes depend on n(1-||phi^n||_L1)^2. The actual assumptions are Condition 2.1: existence of beta_n ->0 with a uniform L2 bound on beta_n psi^n and vague convergence of beta_n psi^n dt to F. That is a scaling compactness condition, not just criticality, and it does not follow from ||phi^n||_L1 ->1. More concretely, the regime parameter in the theorems is zeta = lim n beta_n^2, not n(1-||phi^n||_L1)^2. The stress-test example is valid: take phi_n(t)=a_n b_n phi(b_n t) with phi an alpha-stable density, alpha in (1/2,1), a_n=1-n^{-2/3}, b_n=n^{1/(3 alpha)}, beta_n=n^{-1/3}. This satisfies Condition 2.1 and beta_n mu_0^n -> a, but n beta_n^2 -> infinity while n(1-||phi^n||_1)^2 ->0. So the abstract would call it the synchronized regime; Theorem 3.1 puts it in the extinction regime. That is a real contradiction between the packaging and the content, not a nitpick. The authors either need to restrict to beta_n comparable to 1-||phi^n||_1, or rewrite the abstract and introduction around n beta_n^2 and state that the L1-gap intuition only holds in the classical scaling.\n\nOther soft spots are minor by comparison. Proposition 5.1, which drives the Cox-process representation, has its proof omitted and is only described as a direct consequence; it should be supplied. The L2 condition in Condition 2.1 is admittedly not fully characterized, but the paper says so honestly and gives a sufficient condition in the univariate case.\n\nBottom line: conditional on Condition 2.1, the results appear correct and the mean-field trichotomy is a genuine advance. The abstract and the prominence of the L1-gap phase diagram need substantial revision, but this deserves a serious referee and, after fixing the presentation, would be a good paper.","headline":"A solid conditional theorem with a genuinely new mean-field trichotomy, but the abstract's L1-based phase diagram is not supported by the theorem's zeta=lim n beta_n^2 and needs major revision.","tokens_in":29649,"tokens_out":6048,"would_cite":true,"duration_ms":60625,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60G55","60G22","60F17","60G57"],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-critical Hawkes scaling limits are affine stochastic Volterra diffusions; mean-field particle systems have three regimes set by $n(1-\\|\\phi^n\\|_{L^1})^2$.","keywords":["Hawkes process","mean-field limit","scaling limit","propagation of chaos","interacting particle system","affine stochastic Volterra equation","nearly unstable Hawkes process","resolvent of the second kind"],"falsifier":"Take a family of kernels with $\\|\\phi^n\\|_{L^1}=1-1/n$ whose mass is concentrated on intervals of width $\\varepsilon_n\\to0$, and check whether any $\\beta_n\\to0$ keeps $\\sup_n\\beta_n\\|\\psi^n\\|_{L^2_T}<\\infty$ while $\\beta_n\\psi^n(t)\\,dt$ has a non-degenerate vague limit; if only degenerate or unbounded rescalings are possible and the rescaled Hawkes process still converges to a non-affine limit, the blanket claim that $L^1$-criticality forces affine Volterra limits would be refuted.","tokens_in":28389,"feed_emoji":"📈","tokens_out":15072,"duration_ms":151551,"temperature":0.7,"pith_summary":"This paper studies Hawkes processes, event-counting processes whose jump rate is increased by past jumps, when the total self-excitation mass approaches the critical value one. It proves that if the cumulative self-excitation kernel can be rescaled so that it converges to a nontrivial measure with controlled second moments, then every limit point of the rescaled processes is an affine stochastic Volterra diffusion driven by a time-changed Brownian motion. For a network of $n$ exchangeable particles with mean-field interaction, the same argument gives propagation of chaos, and a single number, $n(1-\\|\\phi^n\\|_{L^1})^2$, decides between three macroscopic regimes: complete synchronization, conditionally independent Poisson-like fluctuations, or extinction. The upshot is that near-critical self-excitation produces tractable macroscopic equations rather than arbitrary stochastic limits.","feed_headline":"One number decides near-critical Hawkes fates","feed_subtitle":"A single scaling parameter separates synchronized, independent, and extinct behavior.","key_machinery":"The carrying object is the second-kind resolvent $\\psi^n=\\sum_{k\\ge1}(\\phi^n)^{*k}$, the sum of all convolution powers of the self-exciting kernel, which encodes the total cumulative effect of repeated excitation. The martingale representation $\\lambda^n(t)=\\mu^n+\\|\\psi^n\\|_{L^1_t}\\mu^n+\\psi^n*dM^n(t)$ rewrites the intensity as a deterministic drift plus a stochastic convolution against the compensated point process. The vague convergence of $\\beta_n\\psi^n(t)\\,dt$ fixes the kernel $f$ in the limiting Volterra equation, while the uniform bound on $\\beta_n\\|\\psi^n\\|_{L^2_T}$ supplies the tightness estimates that force the limit to be continuous and the martingale to converge to a time-changed Brownian motion. In the mean-field proof, an auxiliary particle system built from the same Poisson measures but with the first $K$ coordinates removed is shown to be asymptotically equivalent to the original system, which makes the first $K$ particles conditionally independent Cox processes time-changed by the aggregate compensator.","core_discovery":"The central claim is that near-criticality is a general source of affine stochastic Volterra limits. Under Condition 2.1, meaning there is $\\beta_n\\to0$ with $\\sup_n \\beta_n\\|\\psi^n_{ij}\\|_{L^2_T}<\\infty$ and $\\beta_n\\psi^n_{ij}(t)\\,dt$ converging vaguely to $F_{ij}(dt)$, together with $\\beta_n\\mu^n\\to a$, the rescaled processes $(\\beta_n^2\\Lambda^n,\\beta_n^2N^n,\\beta_nM^n)$ are C-tight, so all limit points are continuous, and every limit $(X,X,Z)$ satisfies $X(t)=\\|F\\|_{L^1_t}\\cdot a+f*Z(t)$ with $Z_i=B_i\\circ X_i$ for a Brownian motion $B$. In the mean-field particle system, the aggregate process is itself a univariate Hawkes process, and the limiting law of any fixed block of $K$ particles is governed by $\\zeta=\\lim n\\beta_n^2$: if $\\zeta=0$, $X_i=X$ and $Z_i=W_i\\circ X$; if $0<\\zeta<\\infty$, $X_i=\\zeta N_i^\\circ(X/\\zeta)$ and $Z_i=\\sqrt{\\zeta}\\,\\widetilde N_i^\\circ(X/\\zeta)$ for independent unit-rate Poisson processes $N_i^\\circ$; and if $\\zeta=\\infty$, $X_i=Z_i=0$.","pith_inferences":["A testable consequence is that $n(1-\\|\\phi^n\\|_{L^1})^2$ acts as an order parameter: for a fixed kernel family, simulations should cross from synchronized to conditionally independent to extinct behavior as this single number is swept.","Because the univariate characterization runs through the Bernstein-function representation, the possible limiting kernels are exactly those with Laplace transform $1/\\Phi$ for a Bernstein function $\\Phi$, so path regularity could be read off from the corresponding L\\'evy triplet rather than from the original kernels.","The auxiliary-coupling method suggests a broader principle: in near-critical mean-field systems, conditional independence of a finite block from the aggregate is structural and may persist for nonlinear intensities satisfying an analogous resolvent condition.","The extinction regime warns that the chosen rescaling can hide all idiosyncratic noise; observing residual fluctuations would require a different normalization."],"forward_implications":["Under Condition 2.1, every weak limit of the rescaled Hawkes compensator and counting process is an affine stochastic Volterra diffusion, so macroscopic equations of square-root or fractional type arise from microscopic self-excitation.","At $\\zeta=0$ the mean-field network synchronizes: each particle has the same limit $X$ as the aggregate, and its noise is a Brownian motion time-changed by $X$.","At $0<\\zeta<\\infty$ the particles are conditionally independent Poisson processes time-changed by $X/\\zeta$, a propagation of chaos toward a stochastic intensity rather than a deterministic one.","At $\\zeta=\\infty$ the rescaled particles vanish: both the counting process and its martingale part converge to zero.","The empirical measures of the particles converge to the conditional law of a single particle given the aggregate, so the three regimes describe the full distributional behavior of a large network."],"supporting_citations":[{"why":"Supplies the original light-tailed nearly unstable scaling limit whose method the paper extends to general kernels.","marker":"[37]"},{"why":"Supplies the heavy-tailed analogue and the fractional-kernel limit that the new characterization must reproduce.","marker":"[38]"},{"why":"Establishes propagation of chaos for stable mean-field Hawkes systems, the framework the paper generalizes to near-criticality.","marker":"[19]"},{"why":"Defines the affine Volterra process class in which the limiting equations live.","marker":"[2]"},{"why":"Gives the resolvent-based martingale representation of Hawkes intensities used throughout the proofs.","marker":"[4]"},{"why":"Provides the C-tightness criteria, Skorokhod convergence tools, and martingale limit theorems used in the proofs.","marker":"[36]"},{"why":"Supplies the martingale representation theorem and Poisson representation used to characterize limits and build the mean-field coupling.","marker":"[35]"},{"why":"Gives the Bernstein-function theory used to characterize all possible univariate limit kernels through their Laplace transforms.","marker":"[49]"},{"why":"Yields absolute continuity of the limiting measure, so the vague limit has the density appearing in the Volterra equation.","marker":"[42]"}],"fun_headline_variants":["Near-critical Hawkes: three fates from one limit","Mean-field Hawkes: three regimes from near-criticality","Nearly unstable Hawkes: synchronized, independent, extinct","Hawkes near criticality: affine Volterra limits","One limit parameter splits Hawkes behavior"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the cumulative self-excitation kernel can be rescaled to converge to a nontrivial limit measure while its rescaled squared size stays uniformly bounded; merely requiring the original kernel mass to approach one does not guarantee this.","fun_headline_variants_meta":{"raw":{"variants":["Near-critical Hawkes: three fates from one limit","Mean-field Hawkes: three regimes from near-criticality","Nearly unstable Hawkes: synchronized, independent, extinct","Hawkes near criticality: affine Volterra limits","One limit parameter splits Hawkes behavior"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00061,"raw_usage":{"total_tokens":2850,"prompt_tokens":970,"completion_tokens":1880,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":1805}},"tokens_in":586,"tokens_out":1880,"duration_ms":17119,"temperature":1.0,"reasoning_tokens":1805,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T18:01:26.687451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a family of kernels with $\\|\\phi^n\\|_{L^1}=1-1/n$ whose mass is concentrated on intervals of width $\\varepsilon_n\\to0$, and check whether any $\\beta_n\\to0$ keeps $\\sup_n\\beta_n\\|\\psi^n\\|_{L^2_T}<\\infty$ while $\\beta_n\\psi^n(t)\\,dt$ has a non-degenerate vague limit; if only degenerate or unbounded rescalings are possible and the rescaled Hawkes process still converges to a non-affine limit, the blanket claim that $L^1$-criticality forces affine Volterra limits would be refuted.","supporting_citations":[{"cited_title":"Jaisson and M","cited_arxiv_id":null,"evidence_quote":"Supplies the original light-tailed nearly unstable scaling limit whose method the paper extends to general kernels."},{"cited_title":"Jaisson and M","cited_arxiv_id":null,"evidence_quote":"Supplies the heavy-tailed analogue and the fractional-kernel limit that the new characterization must reproduce."},{"cited_title":"Delattre, N","cited_arxiv_id":null,"evidence_quote":"Establishes propagation of chaos for stable mean-field Hawkes systems, the framework the paper generalizes to near-criticality."},{"cited_title":"Abi Jaber, M","cited_arxiv_id":null,"evidence_quote":"Defines the affine Volterra process class in which the limiting equations live."},{"cited_title":"Bacry, S","cited_arxiv_id":null,"evidence_quote":"Gives the resolvent-based martingale representation of Hawkes intensities used throughout the proofs."},{"cited_title":"Jacod and A","cited_arxiv_id":null,"evidence_quote":"Provides the C-tightness criteria, Skorokhod convergence tools, and martingale limit theorems used in the proofs."},{"cited_title":"Ikeda and S","cited_arxiv_id":null,"evidence_quote":"Supplies the martingale representation theorem and Poisson representation used to characterize limits and build the mean-field coupling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Bernstein-function theory used to characterize all possible univariate limit kernels through their Laplace transforms."},{"cited_title":"Meyer and W.A","cited_arxiv_id":null,"evidence_quote":"Yields absolute continuity of the limiting measure, so the vague limit has the density appearing in the Volterra equation."}],"review_version":1}