{"id":"4529728b-df2d-4e1a-8b66-e9c96e28b03f","arxiv_id":"2411.16376","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For subordinators whose Laplace exponents admit power expansions, the paper proves higher-order correction terms to the Dynkin-Lamperti arcsine law and matching expansions of potential densities.","lead":"This mathematics paper derives higher-order correction terms for the arcsine limit law that describes where a subordinator sits just before it first jumps above a fixed level. The results provide explicit error rates and next-order densities for the Dynkin-Lamperti theorem, with worked examples for stable and geometric stable subordinators.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the expansion theorems are internally sound, and the flagged condition (B) is explicitly verified in every worked example.","rationale":"The reader identified condition (B) as the weakest assumption, and I agree that it is the least immediately transparent hypothesis. However, after checking the proofs, (B) is not an unjustified step: it is exactly the integrability needed for the Fourier-inversion tail, and it is verified in all examples. I therefore do not find a load-bearing concern that would change the verdict. The minor boundary cases alpha=0 and alpha=1 in the statement of Theorem 3.4 do not affect the main arcsine result. The paper's conditional theorems are rigorous, the examples are concrete, and the claimed error rates follow from the stated hypotheses.","tokens_in":15138,"tokens_out":42001,"duration_ms":366047,"concrete_test":"Derive a sectorial lower bound |Phi(lambda+i theta)| >= c(lambda^alpha + |theta|^alpha) from the Bernstein representation for any subordinator satisfying (A) with 0<alpha<1; if such a bound holds, condition (B) follows for every N>1-alpha and the theorem can be simplified to condition (A) alone.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof of Theorem 3.1, the passage to the Dynkin–Lamperti density in Theorem 3.4, and the four example families. The weakest hypothesis is indeed the uniform tail-integrability condition (B), but it is used exactly where claimed: Lemma 4.1 needs the vertical-line integrability, (4.6) uses (B) for the O(x^{-N}) tail, and (4.7) uses it to control the tail of the difference. Condition (B) is not derived from (A), but it is checked directly in Examples 5.1 and 5.6 with explicit rational and polynomial bounds, and I found no hidden error in those checks. The short-range Theorem 3.6 needs no analogue of (B) because the expansion at infinity makes the relevant integral absolutely convergent. The alpha=0 and alpha=1 boundary values in Theorem 3.4 are degenerate for the arcsine limit, but they do not affect the claimed expansions in the arcsine range 0<alpha<1. I find no circularity, no missing hypothesis, and no step where the central claim overreaches the proof. The only genuine limitation is that (B) is not shown to be automatic in the arcsine range, which weakens the presentation but not the correctness of the theorems as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves higher-order asymptotic expansions of the potential density of a killed subordinator, both at infinity and at zero, under analytic conditions on the Laplace exponent, and then derives corresponding higher-order corrections to the Dynkin--Lamperti arcsine limit for X_{T(s)-}/s. The main abstract results are Theorem 3.1 (long-range potential density expansion) and Theorem 3.6 (short-range version), with Theorems 3.4 and 3.9 giving uniform expansions of the density of X_{T(s)-}/s on compact subintervals of (0,1). The abstract results are applied to several explicit families: sums of stable subordinators, geometric stable subordinators, and mixed cases. The proof is based on a Fourier inversion representation of the potential density (Lemma 4.1) and on control of the remainder through the assumptions (A)/(B) and (A').","tokens_in":15306,"tokens_out":13172,"duration_ms":116717,"significance":"If correct, this is a meaningful quantitative refinement of the classical Dynkin--Lamperti theorem: the paper gives explicit next-order corrections to the arcsine limit, with coefficients and rates read off from the Laurent expansion of 1/Phi. The proofs are complete and self-contained, and the key Fourier-inversion step is cleanly isolated in Lemma 4.1. I particularly appreciate that the coefficients c_k and exponents alpha_k are outputs of the expansion of 1/Phi rather than fitted parameters, and that the uniform integrability condition (B) is verified by explicit bounds in every worked example. The main limitation is that condition (B) is not shown to follow from the regular-variation hypotheses of the Dynkin--Lamperti theorem; it is a separate analytic condition that must be checked case by case. This limits the scope of the abstract theorems but does not affect their correctness, since the examples do verify the condition.","major_comments":[],"minor_comments":[{"comment":"The symbol '/BD{...}' (for instance '/BD{alpha>0}' and '/BD{1<=k+ell<=n}') is never defined; it appears to denote an indicator function. Please replace it with standard indicator notation and define it at first use.","section":"Theorems 3.4, 3.9 and Examples 5.2, 5.4, 5.7"},{"comment":"The term 'special Bernstein function' is used without definition; citing [SV06, BBK+09, SSV12] is acceptable in a remark, but a one-line definition would make the remark self-contained.","section":"Remark 2.3"},{"comment":"The assertion that condition (4.1) is satisfied for every lambda with N=1 is stated without proof; it follows from (A') and the stated decay of R, but a short justification would improve readability.","section":"Proof of Theorem 3.6"}],"recommendation":"accept","confidential_remarks":"In my assessment the paper is correct, original, and well within the scope of the journal. The only caveat is that condition (B) is verified case by case rather than derived from regular variation, but this is a limitation of scope and not a correctness issue. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does what it says: it pushes the Dynkin–Lamperti limit theorem from the leading arcsine term to higher-order corrections, and it proves the expansions rigorously. The main theorems (3.1, 3.4, 3.6, 3.9) are new, and the proofs hold up under checking.\n\nWhat is actually new is a general asymptotic expansion of potential densities of killed subordinators under two analytic conditions (A)/(B), and the resulting higher-order corrections to the long-range and short-range arcsine laws. The method is borrowed from operator renewal theory (Terhesiu), but the application here, with explicit Laurent-coefficient expansions and error bounds, is original. The examples are well chosen: sums of stable subordinators, geometric stable, gamma, and a mixed sum. In the stable-sum case the expansion is checked against Kyprianou's explicit Mittag-Leffler formula, which is good external validation. The coefficients in the expansion are outputs of the analysis, not fitted parameters; there is no circularity.\n\nProof quality is high. Lemma 4.1 is a standard Fourier inversion representation, and the error analysis uses exactly the monotonicity of R and condition (B). I spot-checked the Laurent coefficients and the example computations and found no hidden sign errors or missing hypotheses. The paper is honest about where each condition enters.\n\nSoft spots: condition (B) is technical and not derivable from (A); it is verified case by case, and the paper does not discuss how sharp it is or give simpler sufficient conditions. That is a presentation limitation, not a correctness flaw. The short-range theorems use only N=1, which is fine but thinner than the long-range case. The boundary cases alpha=0,1 are excluded from the arcsine interpretation, as they should be. Error rates are probably not optimal, but the paper does not claim they are.\n\nWho this is for: specialists in Levy processes and potential theory, and people working with operator renewal expansions. It deserves a serious referee and should be accepted. I would bring it to a reading group and would cite it if I worked in this area. The stress-test note found no significant objection, and I concur.","headline":"Genuine higher-order expansion theorem for Dynkin–Lamperti, with complete proofs and worked examples; conditions are technical but the claims hold as stated.","tokens_in":15914,"tokens_out":1945,"would_cite":true,"duration_ms":19526,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60F05","41A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the law of a killed subordinator just before crossing a high level has an explicit asymptotic expansion around the arcsine distribution, with correction terms read off from the Laplace exponent near zero.","keywords":["Dynkin–Lamperti theorem","arcsine law","potential density","subordinator","asymptotic expansion","regular variation","Fourier inversion","killed subordinator"],"falsifier":"Take a Laplace exponent $\\Phi$ that is regularly varying at 0 with index $\\alpha$ but whose $N$-th derivative has a non-integrable singularity along vertical lines, so condition (B) fails, and check whether the potential density still has the claimed expansion; if it does, (B) is not necessary. More directly, for $\\Phi(z)=\\log(1+z^\\alpha)$, numerically invert the Laplace transform of the potential density at large $x$ and compare the second-order coefficient to the prediction involving $b_0=1/2$; a mismatch would disprove the expansion.","tokens_in":14862,"feed_emoji":"","tokens_out":6527,"duration_ms":71731,"temperature":0.7,"pith_summary":"The Dynkin–Lamperti theorem says that for a killed subordinator whose Laplace exponent varies regularly at zero with index $\\alpha \\in (0,1)$, the scaled position just before the first passage over a high level converges to the arcsine ($\\beta$$(\\alpha,1-\\alpha)$) distribution. This paper establishes quantitative versions of that limit: under explicit analytic conditions on the Laplace exponent, the density of $X_{T(s)-}/s$ on compact subintervals of $(0,1)$ has an asymptotic expansion whose leading term is the arcsine density and whose next terms are computed from the coefficients of the expansion of $1/\\Phi$. The key technical step is a matching asymptotic expansion of the killed subordinator's potential density at infinity. Two concrete families illustrate the results: sums of stable subordinators, where the corrections are powers of $s$, and geometric stable subordinators, where the corrections are powers of $s$ even though no closed-form potential density is known.","feed_headline":"Arcsine law gains explicit corrections for killed subordinators","feed_subtitle":"New expansion terms sharpen the Dynkin-Lamperti limit, with errors uniform on compact intervals.","key_machinery":"The load-bearing object is the Fourier inversion identity from the potential measure's Laplace transform: for $N \\ge 1$, $\\int_0^\\infty x^N e^{-zx} U(dx) = (-d/dz)^N (1/\\Phi(z))$ for $\\Re z > 0$. Lemma 4.1 inverts this to write the potential density $u(x)$ as an oscillatory integral of $(-d/dz)^N(1/\\Phi)(x^{-1}+i\\theta)$. The expansion of this derivative near $z=0$ (condition (A)) produces the leading power terms, while condition (B), a uniform integrability of the derivative along vertical lines, lets the Fourier inversion integral be truncated with controlled error. Euler's reflection formula then converts the potential-density expansion into the arcsine-density expansion for the overshoot.","core_discovery":"The central discovery is that, under the analytic conditions (A) and (B) of Theorem 3.1, the potential density of a killed subordinator has the expansion $u(x) = \\sum_{k=0}^n x^{-1+\\alpha_k}/(c_k \\Gamma(\\alpha_k)) + O(\\varepsilon_1(x))$ as $x \\to +\\infty$. Theorem 3.4 then transfers this into a higher-order approximation of the density of $X_{T(s)-}/s$ on compact subintervals of $(0,1)$: the leading term is the arcsine density $\\sin(\\pi\\alpha)/\\pi\\, x^{\\alpha-1}(1-x)^{-\\alpha}$, and the next terms are explicit rational combinations of the coefficients $c_k$, the exponents $\\alpha_k$, and the Lévy-tail error $\\varepsilon_2$. Two worked families show the expansion at work: sums of stable subordinators (with corrections in powers of $s$) and geometric stable subordinators (with corrections in powers of $s$ even though no closed-form potential density is known). A short-range analogue as $s \\to 0+$ is included.","pith_inferences":["The same vertical-line Fourier inversion could be iterated to produce expansions for other passage-time functionals, such as the distribution of the overshoot $X_{T(s)}-s$ or the joint law of $(X_{T(s)-},X_{T(s)})$, because the needed Laplace-transform data is the same.","Condition (B) is likely not the weakest possible: the truncation argument only needs a uniform control of the tail integral, so one could try to relax (B) to an integrated tail condition; if that succeeds, the theorem would cover Laplace exponents with slower decay at infinity.","The coefficients $c_k$ entering the arcsine corrections are exactly the coefficients of the Puiseux expansion of $1/\\Phi$ at 0, suggesting that the higher-order arcsine corrections are a fingerprint of the next scales in the Laplace exponent near zero.","A numerical test on the geometric stable subordinator could verify the predicted coefficients $b_k$ (e.g., $b_{-1}=1$, $b_0=1/2$, $b_1=-1/12$) by inverting the Laplace transform at large $x$; a mismatch would point to an error in the expansion."],"forward_implications":["For subordinators satisfying (A) and (B), the arcsine limit law carries explicit, uniform-in-$x$ correction terms on compact intervals of $(0,1)$; the leading error is of order $\\varepsilon_3(s,x)$ as $s \\to \\infty$.","In the sum-of-stable case $\\Phi(z)=C_1 z^\\alpha + C_2 z^\\beta$, the density of $X_{T(s)-}/s$ has an explicit expansion in powers $s^{-k(\\beta-\\alpha)}$, whose coefficients are computable from $C_1$, $C_2$, $\\alpha$, and $\\beta$.","For geometric $\\alpha$-stable subordinators, $\\Phi(z)=\\log(1+z^\\alpha)$, the expansion proceeds in powers $s^{-k\\alpha}$ even though no closed-form potential density is available.","The short-range theorem gives analogous expansions as $s \\to 0+$ when the Laplace exponent varies regularly at infinity, with the roles of the leading and correction exponents reversed.","The potential-density expansion holds for general killed subordinators without any drift assumption, so it can be applied to subordinators for which series representations for potential densities are not known."],"supporting_citations":[{"why":"Supplies the Dynkin–Lamperti theorem and the identity $P(X_{T(s)-}\\in dx)=\\Pi((s-x,\\infty))U(dx)$ that links the overshoot law to the potential measure.","marker":"[Ber96]"},{"why":"Restates the Dynkin–Lamperti theorem and provides Example 5.25 for the sum-of-stables potential density, used as a comparison in the examples.","marker":"[Kyp14]"},{"why":"Provides Karamata's Tauberian theorem and regular-variation facts that connect the Laplace exponent's regular variation to the Lévy tail and the potential measure.","marker":"[BGT89]"},{"why":"Modelled the proof method: estimating the Fourier inversion of the derivative of $1/\\Phi$ via remainder terms and tail truncation.","marker":"[Ter16]"},{"why":"Supplies the Fourier inversion theorem used in Lemma 4.1 to represent the potential density as an integral.","marker":"[Dur19]"},{"why":"Gives the Mittag-Leffler asymptotic expansions used in the examples to verify the abstract theorems in concrete cases.","marker":"[EMOT55]"}],"fun_headline_variants":["Arcsine law gains explicit higher-order terms","Sharper arcsine laws via higher-order expansions","Killed subordinators: explicit arcsine corrections","Beyond the leading arcsine density","New terms refine the arcsine law for killed subordinators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The separate integrability condition (B), requiring the $N$-th derivative of $1/\\Phi$ to be uniformly integrable along vertical lines, must hold for the Fourier-inversion proof to control tails; the Dynkin–Lamperti hypothesis of regular variation alone does not guarantee it.","fun_headline_variants_meta":{"raw":{"variants":["Arcsine law gains explicit higher-order terms","Sharper arcsine laws via higher-order expansions","Killed subordinators: explicit arcsine corrections","Beyond the leading arcsine density","New terms refine the arcsine law for killed subordinators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00064,"raw_usage":{"total_tokens":2878,"prompt_tokens":807,"completion_tokens":2071,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":423,"completion_tokens_details":{"reasoning_tokens":1999}},"tokens_in":423,"tokens_out":2071,"duration_ms":20499,"temperature":1.0,"reasoning_tokens":1999,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:11:56.985225+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a Laplace exponent $\\Phi$ that is regularly varying at 0 with index $\\alpha$ but whose $N$-th derivative has a non-integrable singularity along vertical lines, so condition (B) fails, and check whether the potential density still has the claimed expansion; if it does, (B) is not necessary. More directly, for $\\Phi(z)=\\log(1+z^\\alpha)$, numerically invert the Laplace transform of the potential density at large $x$ and compare the second-order coefficient to the prediction involving $b_0=1/2$; a mismatch would disprove the expansion.","supporting_citations":[],"review_version":1}