Higher order approximations in arcsine laws for subordinators

Toru Sera

Higher order approximations in arcsine laws for subordinators

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Pith's one-line read 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.

desk verdict Genuine higher-order expansion theorem for Dynkin–Lamperti, with complete proofs and worked examples; conditions are technical but the claims hold as stated. read the letter →

arxiv 2411.16376 v1 pith:COH7MFTD submitted 2024-11-25 math.PR

Toru Sera This is my paper · ORCID

classification math.PR MSC 60G5160F0541A60

keywords Dynkin–LampertitheoremarcsinelawpotentialdensitysubordinatorasymptoticexpansionregularvariationFourierinversionkilled

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

What 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.

Proof 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.

Soft 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.

Who 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.

Referee Report

0 major / 3 minor

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').

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.

minor comments (3)

[Theorems 3.4, 3.9 and Examples 5.2, 5.4, 5.7] 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.
[Remark 2.3] 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.
[Proof of Theorem 3.6] 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.

Circularity Check

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No circularity found: the paper derives higher-order expansions from explicit analytic assumptions on the Laplace exponent, with all coefficients and exponents produced by the proof rather than fitted to the target.

full rationale

The derivation chain is self-contained relative to the stated hypotheses. Theorem 3.1 assumes condition (A), which gives a prescribed Laurent-type expansion of (d/dz)^N (1/Phi) near zero with explicit constants c_k and exponents alpha_k, and condition (B), a uniform tail-integrability bound. The proof then uses Lemma 4.1's Fourier inversion representation, substitutes the assumed expansion, and bounds the remainder via (B); the resulting potential density expansion is an output of this computation, not an input. Theorem 3.4 combines this expansion with the identity (2.6) and the explicit tail of the Levy measure to produce the higher-order Dynkin-Lamperti approximation, again as a consequence rather than by construction. No parameter is fitted to the arcsine density or to any target distribution, and no self-citation is used to justify the central premise. The author cites [Ter16] only as a proof technique to imitate, which is methodological provenance, not load-bearing circularity. External checks, such as recovering Kyprianou's explicit formula in Example 5.5 via the Mittag-Leffler representation, provide independent confirmation. Condition (B) is indeed an extra analytic restriction, but it is stated as an assumption, used exactly where claimed in (4.6)-(4.7), and explicitly verified in Examples 5.1 and 5.6 with rational and polynomial bounds. Thus the central claims reduce neither by definition nor by self-citation.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The proof is a derivation from the stated analytic conditions; no free parameters are fitted. The central claim rests on the two abstract conditions (A) and (B), the Fourier inversion representation, and standard Levy process facts. The examples additionally use known asymptotic expansions for Mittag-Leffler functions and explicit Levy measures from the cited literature.

assumptions (8)

standard math Laplace exponent representation (2.2) with killing rate a, drift b, and Levy measure Pi satisfying (2.3); X is not identically zero.
Background from Bertoin and Kyprianou; used throughout to define Phi and the potential measure.
standard math The potential measure U satisfies int e^{-zx} U(dx) = 1/Phi(z) for Re z > 0, hence (2.5) holds by differentiation.
Laplace transform identity for subordinators; central to Lemma 4.1.
standard math Fourier inversion formula for finite measures (Durrett, Theorem 3.3.14) applies to the complex Laplace transform on vertical lines.
Used in Lemma 4.1 to represent the potential density as an inverse Fourier integral.
domain assumption Condition (A) of Theorem 3.1: (d/dz)^N(1/Phi(z) - sum_{k=0}^n 1/(c_k z^{alpha_k})) = O(R(|z|)) as z -> 0 in the right half-plane, with c_k nonzero, alpha_0 in [0,1], alpha_n > -N+1, and R non-increasing with R(t)=o(t^{-N-alpha_n}).
Primary hypothesis of the main theorem; restricts the class of subordinators.
domain assumption Condition (B) of Theorem 3.1: uniform integrability of the N-th derivative of 1/Phi along vertical lines in R\(-r,r).
Controls the tail of the Fourier integral; separately verified in each example.
domain assumption Condition (A') of Theorem 3.6: analogous expansion of Phi'/Phi^2 as z -> infinity with exponents beta_k.
Primary hypothesis of the short-range theorem.
standard math Karamata's Tauberian theorem and the monotone density theorem (BGT89) relate Phi's regular variation to the tail Pi and the potential density in remarks.
Used in Remarks 3.2, 3.5, 3.7, 3.10 to connect the conditions to classical formulations.
standard math Known asymptotics of the Mittag-Leffler function E_{a,b}(-x) from Erdelyi et al., used in Examples 5.5 and 5.7.
Provides the explicit higher-order terms for the sum of two stable subordinators and for geometric stable subordinators.

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Pith. "Pith review of Higher order approximations in arcsine laws for subordinators." pith.science (2026). https://pith.science/paper/COH7MFTD

@misc{pith2026241116376,
  author       = {Pith},
  title        = {Pith review of: Higher order approximations in arcsine laws for subordinators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/COH7MFTD}},
  note         = {Machine review of arXiv:2411.16376}
}

read the original abstract

We establish higher order approximations in the Dynkin--Lamperti theorem, a limit theorem for the distribution of a killed subordinator immediately before its first passage time over a fixed level. For this purpose, we also study asymptotic expansions of potential densities for killed subordinators.

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Potential analysis of stable processes and its extensions , volume 1980 of Lecture Notes in Mathematics

Krzysztof Bogdan, Tomasz Byczkowski, Tadeusz Kulczycki, Michal Ryznar, Renming Song, and Zoran Vondra c ek. Potential analysis of stable processes and its extensions , volume 1980 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2009

work page 1980

[2] [2]

L\'evy processes , volume 121 of Cambridge Tracts in Mathematics

Jean Bertoin. L\'evy processes , volume 121 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1996

1996

[3] [3]

N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular variation , volume 27 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1989

1989

[4] [4]

I. S. Borisov and E. I. Shefer. Convergence rates to the arcsine law. Theory Probab. Appl. , 68(2):175--197, 2023. Translation of Teor. Veroyatn. Primen. 68 (2023), 209--235

work page 2023

[5] [5]

T. Chan, A. E. Kyprianou, and M. Savov. Smoothness of scale functions for spectrally negative L \'evy processes. Probab. Theory Related Fields , 150(3-4):691--708, 2011

work page 2011

[6] [6]

Stein's method of exchangeable pairs for the beta distribution and generalizations

Christian D \"o bler. Stein's method of exchangeable pairs for the beta distribution and generalizations. Electron. J. Probab. , 20:no. 109, 34, 2015

work page 2015

[7] [7]

( N on)differentiability and asymptotics for potential densities of subordinators

Leif D \"o ring and Mladen Savov. ( N on)differentiability and asymptotics for potential densities of subordinators. Electron. J. Probab. , 16:No. 17, 470--503, 2011

work page 2011

[8] [8]

Probability---theory and examples , volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics

Rick Durrett. Probability---theory and examples , volume 49 of Cambridge Series in Statistical and Probabilistic Mathematics . Cambridge University Press, Cambridge, fifth edition, 2019

2019