{"id":"8404cab0-0dfe-460d-a00b-e6023f13a4fe","arxiv_id":"2608.11637","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For centered Lévy processes with subexponential positive jumps, the local probability of a large deviation at any time s up to t is asymptotically s times the Lévy measure of the target interval, uniformly in the level, the interval length, and small spatial shifts.","lead":"This mathematics paper proves a precise formula for the chance that a Lévy process with rare, heavy-tailed jumps lands in a small interval far from its start: the probability is approximately the time s times the corresponding jump-measure mass, uniformly over time, position, and interval length. The result gives a continuous-time, fully uniform version of a known random-walk one-big-jump principle under weaker moment assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.6's transfer to Corollary 2.1 of [2] is asserted, not proved; this is the hinge of Lemma 3.2 and should be verified.","rationale":"The paper is carefully structured and, conditional on Proposition 3.6, the proof of Theorem 1.2 is internally coherent: the reductions to K = 0, the decomposition into M and Z, the bounds on I1-I4, and the extension from delta in [1,2] to delta >= 1 all check out. The single point on which the argument depends without being established is the transfer from the Levy-measure hypotheses to the exact hypotheses of Corollary 2.1 of [2]. The reader identified the same point as the weakest assumption. I agree that this is the most load-bearing step, because Lemma 3.2, and therefore the theorem, collapses if the local big-jump asymptotics of the centered random walk eS_n are not valid with the stated uniformity. However, there is no evidence that the transfer actually fails; Condition 1.1 appears designed to make it work, and the missing piece is a spelled-out verification rather than a demonstrated counterexample. For that reason the appropriate verdict is conditional acceptance pending that verification, not rejection.","tokens_in":24013,"tokens_out":30803,"duration_ms":336853,"concrete_test":"State the hypotheses of Corollary 2.1 of [2] verbatim and verify them for F(x + Delta_delta) = nu(x + m + Delta_delta). In particular, check that Condition 1.1(3) implies (i) F is locally subexponential with truncation sequence n^{1/alpha}, following Lemmas 6.1 and 6.2 of [2] step by step, and (ii) P(Y_1 - m in x - y + Delta_delta) ~ P(Y_1 - m in x + Delta_delta) uniformly for |y| = O(n^{1/alpha}) and x >= theta n. If [2] additionally requires F's tail to be subexponential, derive that from x^alpha nu(x + Delta_delta) in Sd union OR and the almost-decrease property; or exhibit a Levy measure satisfying Condition 1.1 for which the local-subexponentiality condition of [2] fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central mechanism is Lemma 3.2, whose proof reduces the large-jump part Z to the centered random walk eS_n and then uses Proposition 3.6. The entire content of Proposition 3.6 is the claim that Corollary 2.1 of Denisov, Dieker and Shneer [2] applies to the increment distribution F(dx) = P(Y_1 - m in dx), yielding (3.37): sup_{x >= theta n} |P(eS_n in x + Delta_delta) / (n nu(x + Delta_delta)) - 1| -> 0. The only verification offered is E|Y_1 - m|^alpha < infinity. What is not checked is that Condition 1.1(3) implies the two substantive hypotheses of [2]: (i) F is locally subexponential, or satisfies [2]'s local big-jump condition, with truncation sequence n^{1/alpha}; and (ii) the local insensitivity required in [2] holds at scale n^{1/alpha}. Part (ii) is plausible because b(x) >= c x^{1/alpha} and (1.3) covers shifts up to K b(x); part (i) does not follow formally from x^alpha nu(x + Delta_delta) in Sd union OR by any argument given in the paper. Since Proposition 3.6 is then used to control the terms I1 and I3 in Lemma 3.2, and Lemma 3.2 is the whole big-jump decomposition behind Theorem 1.2, this unverified transfer is the most load-bearing step. No contradiction is claimed; the step is simply not demonstrated in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a uniform local large-deviation theorem for one-dimensional centered Lévy processes with subexponential jumps. Under Condition 1.1, Theorem 1.2 asserts that for any θ, δ₀ > 0 and K ≥ 0, \nP(X_s ∈ x − y + Δ_δ) / (s · ν(x + Δ_δ)) → 1\nuniformly in x ≥ θt, |y| ≤ Kb(x), δ ≥ δ₀ and 0 < s ≤ t, as t → ∞. The proof splits the Lévy process into a small-jump martingale M and a large-jump compound-Poisson component Z, represents Z through a Poisson-random-walk identity, and imports uniform local asymptotics for random walks from Denisov et al. [2]. Several auxiliary estimates for the local Lévy measure and for M are proved in Section 3.1, and Lemma 3.2 is the main decomposition step.","tokens_in":24354,"tokens_out":12510,"duration_ms":147193,"significance":"If the result is correct, it is a substantial contribution: it gives a continuous-time analogue of the big-jump local asymptotics of Denisov et al. [2], with extra uniformity in time, level, spatial shift and interval length, under a one-sided moment assumption. The theorem is precisely stated, the assumptions are explicit and there are no fitted parameters. The paper is also honest in flagging that the earlier continuous-time claim in Xu [10] was not rigorously proved. The main proofs are written in a lemma-based structure, and the auxiliary estimates on the Lévy measure in Section 2 are mostly solid. The central concern is the transfer of the random-walk theorem in [2] to the centered walk èS_n in Proposition 3.6, which is load-bearing for Lemma 3.2 and hence for Theorem 1.2.","major_comments":[{"comment":"The proof of (3.35) and (3.36) is the hinge of Lemma 3.2, but the application of Corollary 2.1 of [2] is asserted rather than verified. After setting F(dx) = P(Y_1 − m ∈ dx), the text says that E|Y_1 − m|^α < ∞ and then that “Corollary 2.1 in [2] along with Proposition 2.3 tells that (3.37)”. Corollary 2.1 of [2] has substantive hypotheses: the increment distribution must satisfy a local subexponentiality or local big-jump condition with the appropriate truncation sequence, and the local probabilities must be insensitive at the scale n^{1/α}. Neither hypothesis is demonstrated here. Condition 1.1(3) controls ν(x + Δ_δ), not directly the law of Y_1 − m; the relation between the two requires a separate argument, and the one-sided moment condition E[(X_1^+)^α] < ∞ certainly does not by itself place the two-sided increment distribution of Y_1 − m in the class required by [2]. Since Proposition 3.6 is used to control the terms I_1 and I_3 in the proof of Lemma 3.2, this is a load-bearing gap. Please provide an explicit verification of the hypotheses of [2, Cor. 2.1] for F, or prove (3.37) directly.","section":"Section 3.2, Proposition 3.6"},{"comment":"The extension from δ ∈ [1,2] to δ ∈ [1,∞] is carried out by decomposing the interval Δ_δ into [δ]−1 unit intervals. For δ = ∞, however, the quantity [δ] is undefined and the decomposition becomes an infinite sum. Since the theorem explicitly requires uniformity over δ = ∞, a truncation or monotone-convergence argument should be added to justify passing from finite [δ] to the limiting case. This is a local fix, but it concerns the full statement of the main result.","section":"Section 3, Step 2 of the proof of Theorem 1.2"}],"minor_comments":[{"comment":"The abstract states the denominator as s · P(X_1 ∈ (x,x+δ]), while Theorem 1.2 uses s · ν(x + Δ_δ). These are asymptotically equivalent once the theorem is proved, but the two formulations should be aligned to avoid confusing the reader.","section":"Abstract and Theorem 1.2"},{"comment":"In the estimate for ε(R,t), the text says “by Proposition 3.5 (for small n) and 3.6 (for large n)” without spelling out how the intermediate range is covered; a short sentence explaining the split would make the double limit in R and t fully transparent.","section":"Section 3.2, proof of Lemma 3.2"},{"comment":"The proof invokes Lemma 2.3 of [4] without stating why its assumptions are satisfied. Since this proposition is used in the estimates for I_2 and I_4, it would be helpful to state the relevant condition from [4] explicitly.","section":"Section 2, Proposition 2.4"},{"comment":"There are several small stylistic repetitions (“which follows that”, “suÿices” ligature artifacts) that should be cleaned up before publication, and the notation [δ] should be defined for δ ∈ [1,∞] or restricted to finite δ.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is in scope for a probability journal and the central idea is credible. My major_revision recommendation rests on the unverified transfer in Proposition 3.6 to [2, Cor. 2.1]; if the authors provide the missing verification (or prove (3.37) directly), I would expect the paper to be acceptable. I see no sign of circularity, and the treatment of the earlier non-rigorous claim in Xu [10] is transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Liora -- quick take on arXiv:2608.11637.\n\nThis paper is a solid piece of heavy-tailed probability. The main theorem -- uniform local big-jump asymptotics for centered Lévy processes, with the supremum over time s, level x ≥ θt, shift y up to Kb(x), and interval length δ ≥ δ0 -- is genuinely new and clearly worth having. It upgrades Denisov-Dieker-Shneer from random walks to Lévy processes, and it does so under a weaker one-sided moment assumption, using a scale b that adapts to the process. The proof is organized and mostly written with care: the Lévy-Itô split into small-fluctuation martingale M and compound-Poisson big-jump part Z, the Poissonization of Z, and the interval partition arguments are all standard moves executed cleanly. Propositions 2.2-2.5 and Lemma 3.1 are fine. I don't see post-hoc fitting or circular reasoning; the self-citation to Xu is properly flagged as non-rigorous and not used as support.\n\nThe real soft spot is exactly where the stress-test points: Proposition 3.6. Lemma 3.2 -- the whole big-jump decomposition -- reduces the large-jump part to the centered random walk eS_n and then invokes Corollary 2.1 of Denisov et al. The only verification offered is E|Y1-m|^α < ∞. What is missing is a check that the increment distribution F(dx) = P(Y1 - m ∈ dx) satisfies the local subexponentiality and insensitivity hypotheses that Corollary 2.1 actually needs, at scale n^{1/α}. Condition 1.1(3) gives local heavy-tail and insensitivity statements for the Lévy measure ν(x+Δδ), and F is just a normalized shift of that measure, so the transfer is very plausible; but it is not a one-line consequence unless you know the relevant closure properties of local subexponentiality. The paper does not supply that argument. Since (3.35) controls the I1 and I3 terms in Lemma 3.2, this unverified transfer is load-bearing. I would call it a fixable gap rather than a fatal flaw: the missing step can almost certainly be supplied from Asmussen-Foss-Korshunov or the lemmas in [2], but a referee needs to see it.\n\nMinor: Proposition 2.1 says 'proof omitted' -- it's simple, so just include it. A few other quoted external results have slightly compressed hypothesis checking, but nothing else looks dangerous.\n\nBottom line: this deserves serious peer review. I'd send it out, with a request to fill the Proposition 3.6 transfer and spell out the hypotheses of [2] being verified. If that step is as routine as I suspect, the paper will be a nice addition to the heavy-tailed Lévy literature. Worth a reading-group slot.","headline":"A genuine uniform local big-jump theorem for Lévy processes, with one load-bearing hypothesis transfer in Proposition 3.6 that the authors need to write out.","tokens_in":24870,"tokens_out":3525,"would_cite":true,"duration_ms":36248,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A centered subexponential Lévy process has uniform local large-deviation asymptotics given by the Lévy measure, at every time.","keywords":["large deviation","Lévy process","local probability","subexponentiality","big-jump principle","uniform asymptotics","O-regular variation","heavy tails"],"falsifier":"The claim can be tested on the canonical spectrally positive example ν(dy)=$y^{{-1-α}}$ dy on (1,∞), σ=0, α=3/2, b(t)=C $t^{{1/α}}$: evaluate numerically the ratio in (1.4) over a grid of (s,x,y,δ) and check that its supremum tends to zero; a persistent positive gap would be a counterexample. Independent of numerics, the load-bearing step is Proposition 3.6, so one can also inspect whether Condition 1.1 implies the hypotheses of Corollary 2.1 in [2] for the centered random walk of large jumps; a failure there would break Lemma 3.2 and hence the theorem.","tokens_in":23813,"feed_emoji":"⚡","tokens_out":9455,"duration_ms":99588,"temperature":0.7,"pith_summary":"This paper aims to prove that for a centered, one-dimensional Lévy process whose positive jumps are subexponential and have a finite moment of some order α∈(1,2], the probability of landing in a small interval far to the right is asymptotically the observation time multiplied by the Lévy measure of that same interval. The error is shown to vanish uniformly in all four parameters at once: the time horizon s≤t, the level x≥θt, a spatial perturbation y of size at most Kb(x), and the interval length δ≥δ0, including δ=∞. This matters because heavy-tailed large deviations are not shaped by the bulk of the process but by a single exceptional jump; knowing the local probability at every time and scale at once is the kind of input needed for precise ruin, queueing, and rare-event simulation arguments. The paper claims this occurs under a one-sided moment condition, weaker than the two-sided assumption used in the corresponding random-walk result.","feed_headline":"One big jump governs Levy local large deviations at every time","feed_subtitle":"Error vanishes uniformly over time s, level x, spatial shift y, and interval length: the continuous-time big-jump principle.","key_machinery":"The load-bearing object is the natural-scale function b, an eventually non-decreasing, O-regularly varying function with b(t)=o(t) and b(t)≥const·$t^{{1/α}}$, chosen so typical fluctuations of X_t are of order at most b(t) and the local Lévy mass ν(x+Δδ) is insensitive to shifts of that size. The proof splits X via the Lévy–Itô decomposition into a small-jump part M (Brownian motion plus compensated jumps of size ≤1) and a large-jump compound-Poisson part Z. Z is Poissonized: at rate one, its jumps are i.i.d. copies with law ν restricted to (1,∞), and Z_t has the law of a centered random walk in a Poisson time, so the centered walk of jumps minus their mean enters. The paper transfers the known uniform big-jump asymptotic for such centered random walks (Corollary 2.1 of [2]) to this setting (Proposition 3.6), combines it with exponential tail bounds for M and a logarithmic lower bound for ν(x+Δδ), and finally upgrades uniformity from δ∈[1,2] to all δ≥δ0 by an interval-partition step.","core_discovery":"The central discovery is a uniform local big-jump asymptotic. Under Condition 1.1, for any θ>0, δ0>0, K≥0, $$\\lim_{t\\to\\infty}\\sup_{x\\geq\\$\\theta$ t}\\sup_{|y|\\leq Kb(x)}\\sup_{\\delta\\in[\\delta_0,\\infty]}\\sup_{0