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Infinite divisibility of $\alpha$-Cauchy distributions

T0 review · 0 major / 2 minor · reviewed 2026-05-16 · grok-4.3

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Pith's one-line read The α-Cauchy random variable is infinitely divisible if and only if 1 < α ≤ 2.

desk verdict The paper settles the 2009 question with a clean if-and-only-if result: α-Cauchy is infinitely divisible exactly for 1 < α ≤ 2. read the letter →

arxiv 2512.23164 v3 submitted 2025-12-29 math.PR

Min Wang This is my paper · ORCID
classification math.PR MSC 60E07
keywords infinitedivisibilityalpha-CauchydistributionCauchyLevyprocessescharacteristicfunctionLevy-Khinchineformula
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 paper establishes that the α-Cauchy distribution C_α is infinitely divisible precisely when the parameter satisfies 1 < α ≤ 2. This resolves an open question posed in 2009 about the infinite divisibility of these distributions for α greater than 1. Infinite divisibility is a key property in probability theory because it allows the distribution to serve as the law of increments for Lévy processes and relates to the decomposability into sums of independent random variables. For the boundary case α = 2, the distribution reduces to the standard Cauchy, which was already known to be infinitely divisible and stable. The proof involves analyzing the characteristic function and checking the conditions from the Lévy-Khinchine representation.

What carries the argument

The characteristic function of the α-Cauchy variable C_α, analyzed through the Lévy-Khinchine representation to verify the infinite divisibility criterion.

What would settle it

A numerical check showing that the n-th root of the characteristic function fails to be positive definite for some α > 2, or succeeds for all n when 1 < α ≤ 2.

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Extended reading notes

Core claim

We prove that C_α is infinitely divisible if and only if 1 < α ≤ 2. The particular case C_2 is the standard Cauchy variable, which is infinitely divisible and stable. For α ≠ 2 the infinite divisibility was previously unknown, and the result shows it holds exactly in the interval 1 < α ≤ 2.

Load-bearing premise

The α-Cauchy variable is defined in a way that permits direct examination of its characteristic function against the Lévy-Khinchine criterion for infinite divisibility.

Editorial extensions

If this is right

  • The standard Cauchy distribution (α=2) is recovered as the known infinitely divisible case.
  • For 1 < α < 2, new families of infinitely divisible distributions are identified.
  • For α > 2, the α-Cauchy distribution lacks infinite divisibility, preventing its use in certain Lévy process constructions.
  • This completes the classification of infinite divisibility for the α-Cauchy family.

Reading between the lines

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

  • One could attempt to construct explicit Lévy processes with α-Cauchy marginal distributions for α in (1,2].
  • The result may extend to questions about self-decomposability or other divisibility properties for these distributions.
  • Connections to stable distributions suggest possible limits or approximations when α approaches 2.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The manuscript resolves the 2009 question posed by Yano, Yano and Yor on the infinite divisibility of the α-Cauchy random variable C_α for α > 1. It proves that C_α is infinitely divisible if and only if 1 < α ≤ 2, recovering the known case for α = 2 (standard Cauchy) and showing that the associated Lévy measure takes negative values for α > 2, violating the Lévy-Khinchine conditions.

Significance. This provides a complete characterization of infinite divisibility for the α-Cauchy family, extending the stable case at α = 2. The explicit derivation of the characteristic function (Definition 1.1 and Proposition 2.3) followed by direct substitution into the Lévy-Khinchine representation, with verification of non-negativity and integrability of the Lévy measure for 1 < α ≤ 2, constitutes a clean algebraic-analytic resolution of the open problem.

minor comments (2)
  1. [Definition 1.1] Definition 1.1: the precise normalization or density form of C_α for α ≠ 2 should be stated explicitly alongside the characteristic function to allow immediate verification of the subsequent Lévy measure calculations.
  2. [Proposition 2.3] Proposition 2.3: while the characteristic function is given explicitly, a short remark on the domain of analytic continuation or convergence of the integral representation would aid readers in confirming the substitution step into the Lévy-Khinchine formula.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. The report accurately captures the resolution of the 2009 question of Yano, Yano and Yor.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is direct analytic verification

full rationale

The paper defines the α-Cauchy random variable C_α explicitly, derives its characteristic function in closed form (Definition 1.1, Proposition 2.3), and substitutes the resulting expression into the Lévy-Khinchine formula. It then checks non-negativity and integrability of the implied Lévy measure by direct computation for 1 < α ≤ 2 versus α > 2. All steps are algebraic or analytic identities with no fitted parameters renamed as predictions, no self-referential definitions, and no load-bearing self-citations. The 2009 Yano-Yano-Yor question is external and merely poses the problem; the proof does not rely on any prior result by the same author. The central claim therefore stands on independent analytic content.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper relies on standard background results from probability theory concerning infinitely divisible distributions and characteristic functions; no new free parameters or invented entities are introduced.

assumptions (1)
  • standard math Standard properties of infinitely divisible laws and the Lévy-Khinchine representation hold for the α-Cauchy family when α > 1.
    Invoked to check the infinite-divisibility criterion.

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Cite this review

Pith. "Pith review of Infinite divisibility of $\alpha$-Cauchy distributions." pith.science (2026). https://pith.science/paper/2512.23164

@misc{pith2026251223164,
  author       = {Pith},
  title        = {Pith review of: Infinite divisibility of $\alpha$-Cauchy distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2512.23164}},
  note         = {Machine review of arXiv:2512.23164}
}
abstract

In 2009, Yano, Yano and Yor proposed the question of studying the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ for $\alpha > 1$. The particular case $\mathcal{C}_2$ is the well-known standard Cauchy variable, which is infinitely divisible and indeed stable. For $\alpha \neq 2$, the infinite divisibility of $\mathcal{C}_\alpha$ is previously unknown. In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if and only if $1 < \alpha \leq 2$.

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line.Scand

    Lennart Bondesson. On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line.Scand. Actuar. J., (3-4):225–247, 1987

    work page 1987

  2. [2]

    Springer-Verlag, New York, 1992

    Lennart Bondesson.Generalized gamma convolutions and related classes of distributions and densities, volume 76 ofLecture Notes in Statistics. Springer-Verlag, New York, 1992

    work page 1992

  3. [3]

    HCM property and the half-Cauchy distribution.Probab

    Pierre Bosch. HCM property and the half-Cauchy distribution.Probab. Math. Statist., 35(2):191–200, 2015

    work page 2015

  4. [4]

    Additive properties of the Dufresne laws and their multi- variate extension.J

    Jean-Francois Chamayou and G´ erard Letac. Additive properties of the Dufresne laws and their multi- variate extension.J. Theoret. Probab., 12(4):1045–1066, 1999

    work page 1999

  5. [5]

    On the self-decomposability of the half-Cauchy distribution.J

    Alassane Di´ edhiou. On the self-decomposability of the half-Cauchy distribution.J. Math. Anal. Appl., 220(1):42–64, 1998

    work page 1998

  6. [6]

    The beta product distribution with complex parameters.Comm

    Daniel Dufresne. The beta product distribution with complex parameters.Comm. Statist. Theory Meth- ods, 39(5):837–854, 2010

    work page 2010

  7. [7]

    Daniel Dufresne.Gdistributions and the beta-gamma algebra.Electron. J. Probab., 15:no. 71, 2163– 2199, 2010

    work page 2010

  8. [8]

    Ferreira, R.A.C.and Simon

    T. Ferreira, R.A.C.and Simon. On the log-concavity of the wright function.Constr Approx, 2023

    work page 2023

Show all 26 references
  1. [9]

    Kilbas, Francesco Mainardi, and Sergei Rogosin.Mittag-Leffler functions, related topics and applications

    Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin.Mittag-Leffler functions, related topics and applications. Springer Monographs in Mathematics. Springer, Berlin, second edition, [2020]©2020

    2020

  2. [10]

    On complete monotonicity of three parameter mittag-leffler function.Applicable Analysis and Discrete Mathematics, 15(1):118–128, 2021

    Katarzyna G´ orska, Andrzej Horzela, Ambra Lattanzi, and Tibor K Pog´ any. On complete monotonicity of three parameter mittag-leffler function.Applicable Analysis and Discrete Mathematics, 15(1):118–128, 2021

    2021

  3. [11]

    Moments of gamma type and the Brownian supremum process area.Probab

    Svante Janson. Moments of gamma type and the Brownian supremum process area.Probab. Surv., 7:1–52, 2010

    2010

  4. [12]

    On some new moments of gamma type.Statist

    Tetyana Kadankova, Thomas Simon, and Min Wang. On some new moments of gamma type.Statist. Probab. Lett., 165:108854, 7, 2020

    2020

  5. [13]

    D. B. Karp and E. G. Prilepkina. Completely monotonic gamma ratio and infinitely divisibleH-function of Fox.Comput. Methods Funct. Theory, 16(1):135–153, 2016

    2016

  6. [14]

    Infinite divisibility and variance mixtures of the normal distribution.Ann

    Douglas Kelker. Infinite divisibility and variance mixtures of the normal distribution.Ann. Math. Statist., 42:802–808, 1971

    1971

  7. [15]

    Kristiansen

    Gundorph K. Kristiansen. A proof of Steutel’s conjecture.Ann. Probab., 22(1):442–452, 1994

    1994

  8. [16]

    On the law of homogeneous stable functionals.ESAIM Probab

    Julien Letemplier and Thomas Simon. On the law of homogeneous stable functionals.ESAIM Probab. Stat., 23:82–111, 2019

    2019

  9. [17]

    Wojciech M lotkowski and Karol A. Penson. Probability distributions with binomial moments.Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17(2):1450014, 32, 2014

    2014

  10. [18]

    A. Yu. Popov and A. M. Sedletskiui. Distribution of the zeros of the Mittag-Leffler function.Dokl. Akad. Nauk, 390(2):165–168, 2003

    2003

  11. [19]

    A. Yu. Popov and A. M. Sedletskiui. Distribution of roots of Mittag-Leffler functions.Sovrem. Mat. Fundam. Napravl., 40:3–171, 2011; translation inJ. Math. Sci.(N.Y.) 190, no.2, 209–409, 2013

    2011

  12. [20]

    A. V. Pskhu. On the real zeros of a function of Mittag-Leffler type.Mat. Zametki, 77(4):592–599, 2005

    2005

  13. [21]

    Cambridge University Press, Cambridge, 1999

    Ken-iti Sato.L´ evy processes and infinitely divisible distributions, volume 68 ofCambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author

    1999

  14. [22]

    A. M. Sedletskiui. Nonasymptotic properties of the roots of a function of Mittag-Leffler type.Mat. Zametki, 75(3):405–420, 2004. 12

    2004

  15. [23]

    A probabilistic perspective on feller, pollard and the complete monotonicity of the mittag-leffler function.arXiv preprint arXiv:2301.01466, 2023

    Nomvelo Karabo Sibisi. A probabilistic perspective on feller, pollard and the complete monotonicity of the mittag-leffler function.arXiv preprint arXiv:2301.01466, 2023

    2023

  16. [24]

    Steutel and Klaas van Harn.Infinite divisibility of probability distributions on the real line, volume 259 ofMonographs and Textbooks in Pure and Applied Mathematics

    Fred W. Steutel and Klaas van Harn.Infinite divisibility of probability distributions on the real line, volume 259 ofMonographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 2004

    2004

  17. [25]

    On the laws of first hitting times of points for one-dimensional symmetric stable L´ evy processes

    Kouji Yano, Yuko Yano, and Marc Yor. On the laws of first hitting times of points for one-dimensional symmetric stable L´ evy processes. InS´ eminaire de Probabilit´ es XLII, volume 1979 ofLecture Notes in Math., pages 187–227. Springer, Berlin, 2009

    1979

  18. [26]

    V. M. Zolotarev.One-dimensional stable distributions, volume 65 ofTranslations of Mathematical Mono- graphs. American Mathematical Society, Providence, RI, 1986. Translated from the Russian by H. H. McFaden. School of Mathematics and Statistics, Wuhan University of Technology,...

    1986

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