Pith. sign in

REVIEW 3 minor 37 references

A criterion determines when distribution functions with a dominated continuous singular part belong to the quasi-infinitely divisible class Q.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Download PDF Read on arXiv ↗

A criterion is derived for quasi-infinitely divisible distributions with dominated continuous singular parts, generalizing prior discrete and mixed cases while describing the Lévy-type triplet and solving a decomposition problem.

T0 review reviewed 2026-05-22 challenge →

load-bearing objection This paper gives a criterion for quasi-infinitely divisible distributions when the continuous singular part is dominated by the discrete part and solves the Lindner-Pan-Sato decomposition in that case.

arxiv 2505.01148 v1 submitted 2025-05-02 math.PR

The class boldsymbol{Q} and mixture distributions with dominated continuous singular parts

A. A. Khartov This is my paper
classification math.PR
keywords quasi-infinitely divisible distributionsclass Qcontinuous singular partmixture distributionsLévy-type representationinfinitely divisible distributionsrational infinite divisibilitydecomposition problem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 seeks conditions under which a distribution function F with a possible continuous singular component belongs to the class Q of rationally infinitely divisible laws. This class consists of distributions that factor as the convolution of two infinitely divisible distributions and therefore admit a Lévy-type representation with a signed spectral measure. The authors give an explicit criterion that works when the continuous singular part is dominated by the discrete part. The result extends earlier criteria that covered only discrete laws and mixtures of discrete with absolutely continuous laws. They also characterize the corresponding characteristic triplet and prove that the domination assumption cannot be dropped without further restrictions.

Core claim

Under the assumption that the continuous singular part of a distribution function F is dominated by the discrete part in a certain sense, F belongs to the class Q of rationally infinitely divisible distribution functions if and only if the mixture of its discrete and absolutely continuous parts satisfies the corresponding rational infinite divisibility condition. The Lévy-type representation then has a characteristic triplet that may contain a continuous singular component in the spectral measure. The criterion recovers the known results for purely discrete laws and for discrete-absolutely continuous mixtures as special cases, applies to concrete examples, and solves the decompositionproblem

What carries the argument

The domination condition on the continuous singular part by the discrete part of F, which permits extension of the rational infinite divisibility criterion and produces a Lévy-type representation with signed spectral measure.

Load-bearing premise

The continuous singular part of the distribution function must be dominated by the discrete part in a certain sense.

What would settle it

A concrete distribution function whose continuous singular part is not dominated by its discrete part, yet still factors as the convolution of two infinitely divisible distributions, would show the assumption is not necessary.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The criterion applies to specific mixture distributions with continuous singular components.
  • The decomposition problem of Lindner, Pan and Sato receives a positive solution under the domination assumption.
  • The associated Lévy-type representation may include a continuous singular part in its spectral measure.
  • All earlier criteria for discrete laws and discrete-absolutely continuous mixtures are recovered as special cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar domination conditions could be explored for quasi-infinitely divisible laws in higher dimensions or for non-stationary processes.
  • The necessity result indicates where new boundary cases or additional assumptions would be required to enlarge the class Q further.
  • Applications to stochastic modeling may arise whenever jump and singular continuous behaviors must be combined in a single distribution.
Share X Bluesky LinkedIn Reddit HN

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 / 3 minor

Summary. The manuscript proposes a criterion for a distribution function F with a continuous singular part dominated by its discrete part to belong to the class Q of quasi-infinitely divisible laws. Under this assumption, it constructs the signed spectral measure in the Lévy-type representation, generalizes results of Alexeev-Khartov (discrete case) and Berger-Kutlu (discrete-absolutely continuous mixtures), proves necessity of the domination condition via counterexamples, and applies the criterion to resolve the Lindner-Pan-Sato decomposition problem in this regime.

Significance. If the derivations hold, the work meaningfully extends the theory of quasi-infinitely divisible distributions to include dominated continuous singular components, supplying an explicit characteristic triplet and resolving an open decomposition question. Strengths include the explicit construction of the signed spectral measure via adapted Lévy-Khintchine manipulations and the use of counterexamples to demonstrate necessity of the domination hypothesis; these elements provide concrete, falsifiable tools for further analysis in this area of probability theory.

minor comments (3)
  1. Introduction: the statement of the domination condition is phrased as 'in a certain sense'; a precise measure-theoretic formulation (e.g., absolute continuity with respect to the discrete measure or an explicit Radon-Nikodym bound) should appear already here rather than only in the technical sections.
  2. Section 3, after Theorem 3.1: the construction of the continuous singular part of the signed spectral measure is described at a high level; adding one fully worked numerical example with explicit densities or atoms would clarify how the domination hypothesis is used in the integral representation.
  3. References: the citations to Alexeev-Khartov and Berger-Kutlu lack page numbers or arXiv identifiers; completing these entries would improve traceability of the claimed generalizations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The report accurately captures the manuscript's contributions to the theory of quasi-infinitely divisible distributions under the domination condition for continuous singular parts.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a new criterion for membership in class Q under the stated domination assumption on the continuous singular part, using standard Lévy-Khintchine manipulations for signed measures. It explicitly constructs the spectral measure, proves necessity via counterexamples, and applies the result to examples and the Lindner-Pan-Sato problem. Generalizations of prior results by Alexeev-Khartov and Berger-Kutlu are cited as context but do not form a load-bearing self-citation chain; the central argument rests on independent analytic steps adapted to the new hypothesis rather than reducing to fitted inputs, self-definitions, or unverified prior claims by the same authors. The derivation is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, and invented entities cannot be extracted in detail; the work relies on standard properties of characteristic functions and convolutions.

axioms (1)
  • standard math Standard properties of distribution functions, characteristic functions, and the Lévy-Khinchine representation for infinitely divisible laws.
    Invoked implicitly when defining class Q and the signed spectral measure representation.

reviewed 2026-05-22 · how reviews work

0 comments
Cite this review

Pith. "Pith review of The class $\boldsymbol{Q}$ and mixture distributions with dominated continuous singular parts." pith.science (2026). https://pith.science/paper/2505.01148

@misc{pith2026250501148,
  author       = {Pith},
  title        = {Pith review of: The class $\boldsymbolQ$ and mixture distributions with dominated continuous singular parts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2505.01148}},
  note         = {Machine review of arXiv:2505.01148}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the L\'evy-type representation with a ``signed spectral measure''. This class is a wide natural extension of the fundamental class of infinitely divisible distribution functions and it is actively studied now. We are interested in conditions for a distribution function $F$ to belong to the class $\boldsymbol{Q}$ for the unexplored case, where $F$ may have a continuous singular part. We propose a criterion under the assumption that the continuous singular part of $F$ is dominated by the discrete part in a certain sense. The criterion generalizes the previous results by Alexeev and Khartov for discrete probability laws and the results by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In addition, we describe the characteristic triplet of the corresponding L\'evy-type representation, which may contain some continuous singular part. We also show that the assumption of the dominated continuous singular part cannot be simply omitted or even slightly extended (without some special assumptions). We apply the general criterion to some interesting particular examples. We also positively solve the decomposition problem stated by Lindner, Pan and Sato within the considered case.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    I. A. Alexeev, A. A. Khartov, Spectral representations of characteristic functions of d iscrete probability laws, Bernoulli 29 (2023), 2, 1392–1409

    work page 2023

  2. [2]

    I. A. Alexeev, A. A. Khartov, A criterion and a Crame´ er-Wold device for quasi-infinite div isibility for discrete multivariate probability laws , Electron. J. Probab., 28 (2023), 1–17

    work page 2023

  3. [3]

    I. A. Alexeev, A. A. Khartov, On convergence and compactness in variation with a shift of d iscrete probability laws, Vestnik St.Petersburg University: Mathematics, 54 (2021), issue 3, p. 221–226

    work page 2021

  4. [4]

    Applebaum, L´ evy Processes and Stochastic Calculus, Camb

    D. Applebaum, L´ evy Processes and Stochastic Calculus, Camb. Univ. Press, Cambridge, 2009

    work page 2009

  5. [5]

    Baxter, J

    G. Baxter, J. M. Shapiro, On bounded infinitely divisible random variables , Sankhya: The Indian Journal of Statistics, 22 (1960), No. 3/4, 253–260

    work page 1960

  6. [6]

    Berger, On quasi-infinitely divisible distributions with a point ma ss, Math

    D. Berger, On quasi-infinitely divisible distributions with a point ma ss, Math. Nachr., 292 (2019), 1674–1684

    work page 2019

  7. [7]

    Berger, M

    D. Berger, M. Kutlu, Quasi-infinite divisibility of a class of distributions wit h discrete part, Proc. Amer. Math. Soc. 151 (2023), 5, 2211–2224

    work page 2023

  8. [8]

    Berger, M

    D. Berger, M. Kutlu, A. Linder, On multivariate quasi-infinitely divisible distributions , A Lifetime of Excersions Through Random Walks and L´ evy Processes. A Volu me in Honour of Ron Doney’s 80th Birthday. L. Chaumont, A.E. Kyprianou (eds.), Progress in P robability 78 (2021), Birkh¨ auser, 87–120

    work page 2021

  9. [9]

    Bochner, A theorem on Fourier-Stieltjes integrals, B ull

    S. Bochner, A theorem on Fourier-Stieltjes integrals, B ull. Amer. Math. Soc., 40 (1934), 4, 271–276

    work page 1934

  10. [10]

    Chhaiba, N

    H. Chhaiba, N. Demni, Z. Mouayn, Analysis of generalized negative binomial distributions a ttached to hyperbolic Landau levels , J. Math. Phys., 57 (2016), 7, 072103

    work page 2016

  11. [11]

    I. M. Gelfand, D. A. Raikov, G. E. Shilov, Commutative Normed Rings , New York: Chelsea Publishing Company, 1964. 32

    work page 1964

  12. [12]

    B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables , Cambridge: Addison-Wesley, 1954

    work page 1954

  13. [13]

    Hardy, Combinatorics of Partial Derivatives , Electron

    M. Hardy, Combinatorics of Partial Derivatives , Electron. J. Comb., 13 (2006), R1

    work page 2006

  14. [14]

    Hewitt, H

    E. Hewitt, H. S. Zuckerman, Singular measures with absolutely continuous convolution squares, Proc. Camb. Phil. Soc., 62 (1966), , 399–420

    work page 1966

  15. [15]

    O. S. Ivaˇ sev-Musatov,On coefficients of trigonometric null-series . Izv. Akad. Nauk SSSR, Ser. Mat., 21 (1957), 559–578

    work page 1957

  16. [16]

    Jessen, A

    B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function , Trans. Amer. Math. Soc., 38 (1935), 48–88

    work page 1935

  17. [17]

    A. A. Khartov, Compactness criteria for quasi-infinitely divisible distri butions on the integers , Stat. Probab. Lett., 153 (2019), 1–6

    work page 2019

  18. [18]

    A. A. Khartov, A criterion of quasi-infinite divisibility for discrete law s, Stat. Probab. Lett., 185 (2022), 109436

    work page 2022

  19. [19]

    A. A. Khartov, On weak convergence of quasi-infinitely divisible laws , Pacific Journal of Mathematics, 322 (2023), 2, 341–367

    work page 2023

  20. [20]

    A. A. Khartov, Some criteria of rational-infinite divisibility for probab ility laws , arXiv:2305.14524v2

    work page arXiv

  21. [21]

    A. A. Khartov, On decomposition problem for distribution functions of cla ss Q, arXiv:2412.18915

    work page arXiv

  22. [22]

    A. A. Khartov, I. A. Alexeev, Quasi-infinite divisibility and three-point probability l aws, J. Math. Sci. 268 (2022), 731–738

    work page 2022

  23. [23]

    M. G. Krein, Integral equations on the half-line with a kernel depending o n the difference of the argu- ments, Uspekhi Mat. Nauk, 13 (1958), 5, 3–120 (in Russian)

    work page 1958

  24. [24]

    B. M. Levitan, Almost periodic functions , GITTL, Moskow, 1953 (In Russian)

    work page 1953

  25. [25]

    M. A. Lifshits, Random Processes by Example , World Scientific, Singapore, 2014

    work page 2014

  26. [26]

    Lindner, L

    A. Lindner, L. Pan, K. Sato, On quasi-infinitely divisible distributions , Trans. Amer. Math. Soc., 370 (2018), 8483–8520

    work page 2018

  27. [27]

    Yu. V. Linnik, I. V. Ostrovskii, Decomposition of Random Variables and Vectors , Transl. Math. Monog. 48, AMS, Providence, Rhode Island, 1977

    work page 1977

  28. [28]

    Lukacs, Characteristic Functions, Griffin, London, 1970

    E. Lukacs, Characteristic Functions, Griffin, London, 1970

    work page 1970

  29. [29]

    Nakamura, A complete Riemann zeta distribution and the Riemann hypoth esis, Bernoulli 21 (2015), 1, 604–617

    T. Nakamura, A complete Riemann zeta distribution and the Riemann hypoth esis, Bernoulli 21 (2015), 1, 604–617

    work page 2015

  30. [30]

    Passeggeri, On Quasi-Infinitely Divisible Random Measures

    R. Passeggeri, On Quasi-Infinitely Divisible Random Measures . Bayesian Anal., 18 (2023), 1, 253–286

    work page 2023

  31. [31]

    Sato, L´ evy Processes and Infinitely Divisible Distributions, Camb

    K.-I. Sato, L´ evy Processes and Infinitely Divisible Distributions, Camb. Univ. Press, Cambridge, 1999

    work page 1999

  32. [32]

    Schoutens, L´ evy Processes in Finance: Pricing Financial Derivatives, Wiley, Chichester, 2003

    W. Schoutens, L´ evy Processes in Finance: Pricing Financial Derivatives, Wiley, Chichester, 2003. 33

    work page 2003

  33. [33]

    Yu. A. Shreider, The structure of maximal ideals in rings of measures with conv olution, Sbornik Math., 69 (1950), 2, 297–318 (in Russian)

    work page 1950

  34. [34]

    H. G. Tucker, On a necessary and sufficient condition that an infinitely divi sible distribution be absolutely continuous, Trans. Amer. Math. Soc., 118 (1965), 316–330

    work page 1965

  35. [35]

    Wiener, H

    N. Wiener, H. R. Pitt, On absolutely convergent Fourier-Stieltjes transforms , Duke Math. J., 4 (1938), 420–430

    work page 1938

  36. [36]

    Wiener, A

    N. Wiener, A. Wintner, Fourier-Stieltjes transforms a nd singular infinite convolutions, Amer. J. Math., 60 (1938), 3, 513–522

    work page 1938

  37. [37]

    V. M. Zolotarev, Modern Theory of Summation of Random Variables , Utrecht: VSP, 1997. 34

    work page 1997

This paper was first reviewed by grok-4.3 on May 22, 2026.