Uniform Local Asymptotics for L\'evy Processes with Subexponential Jumps

Hao Wu; Wei Xu

REVIEW 2 major objections 4 minor 300 references

Uniform Local Asymptotics for L\'evy Processes with Subexponential Jumps

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A centered subexponential Lévy process has uniform local large-deviation asymptotics given by the Lévy measure, at every time.

desk verdict 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. read the letter →

arxiv 2608.11637 v1 pith:7LAMDNOE submitted 2026-08-12 math.PR

Hao Wu , Wei Xu This is my paper · ORCID

classification math.PR MSC 60G5160F10

keywords largedeviationLévyprocesslocalprobabilitysubexponentialitybig-jumpprincipleuniformasymptoticsO-regularvariationheavytails

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

The central discovery is a uniform local big-jump asymptotic. Under Condition 1.1, for any θ>0, δ0>0, K≥0, $$\lim_{t\to\infty}\sup_{x\geq\$\theta$ t}\sup_{|y|\leq Kb(x)}\sup_{\delta\in[\delta_0,\infty]}\sup_{0<s\leq t}\left|\frac{\mathbf P(X_s\in x-y+\Delta_\delta)}{s\,\nu(x+\Delta_\delta)}-1\right|=0,$$ where $\Delta_\delta=(0,\delta]$. In words: whenever a large positive displacement of order t is observed at any time s≤t, its local probability is, after dividing by s, the Lévy measure of the corresponding interval; fluctuations of the process away from the big jump matter only through the scale b(x) and vanish in relative terms. The uniformity in δ extends to δ=∞, so tail and local statements are recovered as endpoints of one family. The authors present this as a continuous-time and fully uniform analogue of the random-walk big-jump result of [2], obtained under a one-sided rather than two-sided moment assumption.

Load-bearing premise

The whole proof leans on the claim that the random walk formed by the centered large jumps obeys the same uniform approximation already proved for random walks in the cited 2008 paper; the transfer of that approximation to the present setting is asserted rather than checked.

Editorial extensions

If this is right

For any centered subexponential Lévy process satisfying Condition 1.1, P(X_s∈x−y+Δδ) ∼ s·ν(x+Δδ) holds simultaneously over s∈(0,t], x≥θt, |y|≤Kb(x), δ≥δ0, with relative error going to zero.
The endpoint δ=∞ is included, so the local statement contains the uniform tail relation P(X_s∈(x−y,∞)) ∼ s·ν((x,∞)) as a special case.
The theorem implies P(X_1∈x+Δδ) ∼ ν(x+Δδ) uniformly in δ≥δ0, making the local law of the process at unit time asymptotically identical to the Lévy measure on the large-deviation scale.
The one-big-jump mechanism is made quantitative: conditioning on X_s in a far-right window, one exceptional positive jump carries the displacement while the rest of the path stays within order b(x).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: the same proof strategy should extend to path functionals governed by the same one-big-jump realization, such as the running maximum or the overshoot over a level, because the uniformity in s supplies exactly the control those path statements require.
Editorial: the four-fold uniformity suggests relative-error estimates usable for rare-event simulation—an estimator based on sampling the big jump could have an error bound uniform over the whole large-deviation region, not just at a fixed x.
Editorial: the one-sided moment assumption decouples the positive tail from negative fluctuations; a natural next step is a multi-dimensional or additive-process version where each coordinate carries its own scale b_i(t).
Editorial: if the transfer in Proposition 3.6 is ever found deficient, the theorem might still hold by proving the local-insensitivity and truncation-sequence hypotheses of [2] directly from Condition 1.1 rather than citing them.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Liora -- quick take on arXiv:2608.11637.

This 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.

The 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.

Minor: 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.

Bottom 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.

Referee Report

2 major / 4 minor

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, P(X_s ∈ x − y + Δ_δ) / (s · ν(x + Δ_δ)) → 1 uniformly 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.

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 (2)

[Section 3.2, Proposition 3.6] 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 3, Step 2 of the proof of Theorem 1.2] 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.

minor comments (4)

[Abstract and Theorem 1.2] 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 3.2, proof of Lemma 3.2] 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 2, Proposition 2.4] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.2 is derived from explicit assumptions plus cited external results; the sole self-citation is disclaimed and unused.

full rationale

The derivation chain is self-contained relative to its cited external results. Theorem 1.2 is proved by the Lévy–Itô decomposition into the small-jump martingale M and the compound-Poisson large-jump process Z, with Lemma 3.1 controlling M and Lemma 3.2 providing the uniform local asymptotics for Z. Lemma 3.2 uses the representation Z_t = S_{N_t} - mt, a Poisson-random-time decomposition, and Proposition 3.6, which invokes Corollary 2.1 of Denisov et al. [2] for the centered random walk eS_n; this is an external theorem whose transfer is argued via the finite |Y_1 - m|^alpha moment and the truncation/natural-scale discussion, not assumed as the paper's own conclusion. No parameter is fitted to the quantity being predicted, and the uniformity in s, x, y, and delta is achieved by partitioning delta and by Chebyshev/Chernoff estimates for the Poisson clock and the small-jump component. The only self-citation, Xu [10], is explicitly labeled non-rigorous ('the proof given there is not rigorous') and is not used to support any step, so it creates no circular burden. Whether Proposition 3.6 fully verifies all hypotheses of [2] is a correctness and rigor question, not a circularity one.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper contributes a proof under explicit conditions; it introduces no fitted constants and no new objects. The central claim rests on the standing assumptions in Condition 1.1 and on four external results from [1], [2], and [4]; these are treated as axioms here because they are quoted rather than re-proved.

assumptions (5)

standard math Levy-Ito decomposition and Poisson random measure representation
Section 3 represents X = M + Z and uses independence of M and the compound Poisson part; this is the standard structural fact for Levy processes.
domain assumption Corollary 2.1 of Denisov, Dieker and Shneer (2008)
Proposition 3.6 uses it to obtain uniform local asymptotics for the centered random walk eS_n at levels x >= theta n; the paper relies on this external theorem without reproving it.
domain assumption Corollary 2 and Proposition 4 of Asmussen, Foss and Korshunov (2003)
Proposition 3.5 uses them for local subexponential asymptotics of the uncentered random walk S_n.
domain assumption Lemma 2.3 of Denisov, Vatutin and Wachtel (2014)
Proposition 2.4 uses it to control the lower tail of nu(x + Delta_{delta_0}) by exp(-C x^{1-1/alpha}).
domain assumption Condition 1.1(2)-(3): existence of a natural scale b and local insensitivity of nu at that scale
These are standing hypotheses of the theorem; they are not derived. All later estimates are phrased in terms of b.

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Pith. "Pith review of Uniform Local Asymptotics for L\'evy Processes with Subexponential Jumps." pith.science (2026). https://pith.science/paper/7LAMDNOE

@misc{pith2026260811637,
  author       = {Pith},
  title        = {Pith review of: Uniform Local Asymptotics for L\'evy Processes with Subexponential Jumps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LAMDNOE}},
  note         = {Machine review of arXiv:2608.11637}
}

abstract

This paper is devoted to unifying the uniform local large-deviation asymptotics for a centered L\'evy process $X$ with subexponential jumps. Our results assert that for any $\theta,\delta_0>0$ and $K\geq0$, $$\lim_{t\to\infty}\sup_{x\geq\theta t}\sup_{|y|\leq Kb(x)}\sup_{\delta\in[\delta_0,\infty]}\sup_{0<s\leq t}\bigg|\frac{\mathbf P\big(X_s\in(x-y,x-y+\delta]\big)}{s\cdot\mathbf P\big(X_1\in(x,x+\delta]\big)}-1\bigg|=0,$$ where the natural-scale function $b$ satisfies a polynomial growth condition. This provides a continuous-time and simultaneously uniform analogue of the results of Denisov et al. [Ann. Probab., 2008], while being established under a weaker moment assumption.

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Reviewed August 16, 2026 · model on record in the stance chip above.

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