Broadly discrete stable distributions

F. William Townes

REVIEW 3 major objections 5 minor 1 cited by

Broadly discrete stable distributions

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Discrete stable laws now cover all tail indices from 0 to 2

desk verdict Promising characterization, but the uniqueness proof has a real gap: the rho-dependent shift is treated as constant, so Theorem 3.1 isn't fully proved as written. read the letter →

arxiv 2509.05497 v1 pith:RXB5SUZM submitted 2025-09-05 math.ST math.PRstat.TH

F. William Townes This is my paper · ORCID

classification math.STmath.PRstat.TH MSC 60E0762E1060E10

keywords discretestabledistributionsmixedPoisson-stablefamilybinomialthinningPoissontranslationinfinitedivisibilitySibuyadistributionself-decomposabilitycountdata

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

Stable distributions are central to probability theory but are continuous and cannot directly model counts. The paper broadens the discrete stable family by replacing scaling with binomial thinning and location shifts with Poisson translation, and proves that such laws exist for every tail index α in (0,2], not only α≤1. Its main theorem gives a closed probability generating function for the whole family and shows it coincides with the mixed Poisson-stable family, with the Hermite distribution at α=2. The paper further shows these distributions are discretely infinitely divisible, gives a compound Poisson representation through a broad Sibuya summand, and identifies when they are discretely self-decomposable and unimodal.

What carries the argument

The key mechanism is the generating-function equation for discrete stability, built from two count-variable operations: dilation/binomial thinning, a◦X with PGF G(1+a(z−1);X), and Poisson translation, X⊕μ with PGF G(z;X)exp(μ(z−1)). Imposing invariance under these operations forces the PGF to the exponential form above, from which the mixed Poisson-stable identification, the compound Poisson representation via the broad Sibuya distribution, and the self-decomposability conditions all follow.

What would settle it

Solve the functional equation for discrete stability under a distribution whose support is truncated at a finite upper bound and check for a solution with α>2; the theorem asserts none exists, so even a numerically stable PGF solution outside the claimed exponential form would refute the uniqueness part. Alternatively, check the power-series coefficients of the PGF obtained from differential equation (19) for α>2—any negative coefficient would settle impossibility.

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Extended reading notes

Core claim

Theorem 3.1 characterizes discrete stability: a count variable has a discrete stable distribution if and only if its probability generating function is exp((z−1)δ+γ(1−z)^α) when α≠1, or exp((z−1)δ+γ(1−z)log(1−z)) when α=1, with α∈(0,2], γ<0 for α<1, γ≥0 for α=1, γ>0 for α>1, and δ≥αγ. Strict discrete stability is the special cases δ=0 for α<1 and γ=0 for α=1. Corollary 3.1.1 identifies this exact family with the mixed Poisson-stable family, so the previously known strict discrete stable laws are the α≤1 part and α∈(1,2] is new, ending at the Hermite distribution for α=2.

Load-bearing premise

The uniqueness proof treats the translation parameter δ as fixed while taking a limit in the thinning parameter ρ, even though the displayed definition of δ depends on ρ; without a regularity condition ensuring δ stays constant, the limit argument for the PGF does not go through.

Editorial extensions

If this is right

The discrete stable family is now defined for all α∈(0,2], covering heavy-tailed count data (α<1) and light-tailed count data (α>1) in one family.
Every discrete stable law is a mixed Poisson-stable law, so existing computational tools for Poisson-stable mixtures can be applied to the full range of parameters.
Every discrete stable law is discretely infinitely divisible and can be represented as a compound Poisson sum of broad Sibuya variables, giving a jump interpretation.
Under δ≥α^2γ for α≠1 (or δ≥2γ for α=1), the laws are discretely self-decomposable and unimodal; the remaining parameter region exhibits multimodality.
The α=2 endpoint reproduces the Hermite distribution, connecting the general family to classical Poisson–Gaussian count models.

Reading between the lines

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

The paper leaves open the conjecture that every real-valued mixing distribution with a completely monotone bilateral Laplace transform on [0,1] yields a discretely infinitely divisible Poisson mixture; proving this would extend compound-Poisson representation well beyond the stable family.
Discrete stability suggests a testable scale-invariance property for count time series: thinning observed counts by different factors plus Poisson noise should preserve distributional shape within the family.
At α=2 the broad Sibuya summand becomes a two-point distribution, so the family interpolates cleanly between light-tailed Hermite-type counts and heavy-tailed α<1 counts, which may guide parameter estimation for real count data.

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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

Here's the short version: the paper has a genuinely good idea — replacing the location shift in broad stability with Poisson translation to get a discrete stable family over the full α∈(0,2] range, and the claimed family matches the mixed Poisson-stable PGFs from the author's earlier work. The forward direction and the compound Poisson representation are clean and useful. But the uniqueness proof in Appendix A.2.2 has a serious gap: δ is defined from the translation parameter μ_ρ, so it depends on ρ, yet the proof treats it as a constant when passing to the limit that gives equation (19). Without controlling that ρ-dependence, the differential equation does not follow. This gap is load-bearing for Theorem 3.1. There's also a false step in the same proof: Var[X]=E[X] does not imply Poisson (e.g., a mixture with mass 1/2 at 0 and 1/2 at 2 has mean 1, variance 1). The conclusion might still be true, but the written argument fails.

What's actually new and good: the definition of discrete stability with Poisson translation is a natural extension of Steutel–van Harn's strict version and appears to be new. The forward direction (A.1) is correct. The compound Poisson representation with the bSib summand (Prop 4.1, Thm 4.1) checks out and gives a useful construction for count data. The discrete self-decomposability analysis (Prop 4.2) is careful and plausible. The paper also honestly flags the conjecture about real-valued ID mixing distributions.

Soft spots: the uniqueness proof issues are not minor. The α=1 case has the same problem with γ depending on ρ. Also, existence for α∈(1,2] is imported from the author's own Proposition 3.1 in [18], which is not proved here; a referee will need to verify that or ask for a self-contained proof. The parameter constraints δ≥αγ and the sign constraints on γ are partly derived from the broken limit step, so they inherit the fragility. That said, this looks fixable: the claimed family is well-motivated and the representations are independently checkable.

Who should read it: people working on count distributions, discrete infinite divisibility, or Poisson mixtures. It could be a standard reference once the proof is repaired. Right now I wouldn't cite it as an established theorem, but I would send it to a serious referee and ask that referee to focus on A.2.2. Reading group: maybe, as a case study of a proof gap.

Referee Report

3 major / 5 minor

Summary. The paper introduces "broadly discrete stable" distributions by modifying the classical stability equation: scaling is replaced by binomial thinning and location shifts by Poisson translation. Theorem 3.1 claims that a count distribution is discrete stable iff its PGF is exp((z-1)δ+γ(1-z)^α) for α≠1, or exp((z-1)δ+γ(1-z)log(1-z)) for α=1, with α∈(0,2] and parameter constraints (10). Corollary 3.1.1 identifies this family with the mixed Poisson-stable family of the author's prior work [18]. Theorem 4.1 gives a compound Poisson representation with a new "broad Sibuya" summand distribution, implying discrete infinite divisibility. Proposition 4.2 gives sufficient and necessary conditions for discrete self-decomposability. The forward direction and the compound Poisson representation are algebraically clean; the main difficulty lies in the uniqueness proof in Appendix A.

Significance. If the characterization is correct, it completes the discrete analog of stability initiated by Steutel and van Harn, extends the family to all α∈(0,2] (including the Hermite case at α=2), and connects it to the mixed Poisson-stable family with real-valued mixing distributions. The broad Sibuya distribution and the explicit compound Poisson representation are useful and novel. The paper also proposes a concrete, falsifiable PGF family suitable for count-data modeling. However, the uniqueness proof has a load-bearing gap involving a ρ-dependent shift parameter, and one step uses a false implication about Poisson distributions. The central claim is therefore not established as written, although it may be repairable.

major comments (3)

[Appendix A.2.2 and A.2.3] The uniqueness proof defines δ_ρ = μ_ρ ((1-ρ^α)^{1/α}-(1-ρ))^{-1}, making δ explicitly ρ-dependent. The subsequent limit passage with h=(1-ρ)(1-z) treats f(z)=e^{δ(1-z)} as if δ were a fixed constant. The same problem occurs in A.2.3 for γ_ρ defined via μ_ρ. Without a proof that δ_ρ (resp. γ_ρ) is independent of ρ, or at least convergent with controlled error, the limiting equations (19) and (23) do not follow. This is the uniqueness direction of Theorem 3.1, so the characterization is unsupported at this step.
[Appendix A.2.2, exclusion of α>2] The proof states that for α>2, Var[X]=E[X] implies X must be Poisson. This implication is false: for example, P(X=0)=P(X=2)=1/2 has Var[X]=E[X]=1 but is not Poisson. Since this step is used to rule out α>2, a different argument (e.g., a direct factorial-cumulant or coefficient comparison) is needed.
[Theorem 3.1 and Proposition 3.1] Theorem 3.1 asserts that the parameter constraints in (10) are sufficient for the PGF in (9) to be a valid count distribution, but the proof in A.2 only derives these constraints as necessary conditions. The forward direction A.1 begins by assuming the PGF exists. The sufficiency of (10) is effectively delegated to Proposition 3.1, which in turn cites the author's prior result [18] without proof. Since the equivalence with the mixed Poisson-stable family is a central claim, the proof should either establish absolute monotonicity of (9) under (10) directly or explicitly state the exact imported content of [18].

minor comments (5)

[Section 4.2] The displayed definition after Eq. (15) is malformed: "Define 1− ρ/(1−ρ) logρ" should presumably be f(ρ)=1−ρ logρ/(1−ρ). Please correct the formula and the surrounding sentence.
[Appendix A.2.2] The sentence "if α≤0 then lim_{z↑1} G(z)>1" is not correct for all γ<0; the limit is e^γ, which is not necessarily greater than 1. The intended point is that the limit is not 1 unless γ=0.
[Theorem 3.1] The notation DS(α,γ,δ) is used in the theorem statement before it is formally defined. Please define it explicitly, or state that it denotes the family with PGF (9).
[References] Proposition 3.1 and Proposition 2.1 rely on the author's prior preprint [18]. If this item is not yet peer-reviewed, please include a version identifier or a short appendix outlining the proof of the real-valued mixing Poisson construction.
[Figure 1] The six panels in Figure 1 are small and the parameter labels are cramped; larger panels or separate rows would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the DS characterization is proved from Definition 3.4, and the only self-citation ([18]) supplies independent existence content; the Appendix A.2.2 regularity gap is a correctness issue, not circularity.

full rationale

Theorem 3.1's uniqueness direction starts from the defining functional equation (Definition 3.4, Eqs. 5-6) and derives the PGF form (Eq. 9) by a limiting argument; it does not assume Eq. 9 as an input. The sufficiency direction verifies that Eq. 9 satisfies Definition 3.4. Thus the central characterization is self-contained with respect to the paper's own definition. The advertised equivalence to the mixed Poisson-stable family (Corollary 3.1.1) uses Proposition 3.1, which is quoted from the author's prior work [18] without proof; however, [18] is an external result whose assumptions (complete monotonicity of a BLT on [0,1]) do not include discrete stability, so this self-citation is not circular in the sense of reducing the target result to itself. It is used for an interpretive corollary rather than for the derivation of Theorem 3.1. The main concern is the Appendix A.2.2 limit: δ is defined as μ((1-ρ^α)^{1/α}-(1-ρ))^{-1}, which depends on ρ, yet the limit producing Eq. (19) treats δ as a single constant; no proof of constancy or convergence is supplied. This is an omitted regularity proof (and the subsequent claim that Var[X]=E[X] forces Poisson is false), but it is a correctness gap in the derivation, not a circular reduction of the conclusion to the assumptions. Accordingly the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper introduces no fitted parameters. The load-bearing assumptions are standard PGF and stable-law results, the author's prior real-valued Poisson mixture existence result, and an unstated regularity condition in the uniqueness proof.

assumptions (5)

domain assumption Proposition 3.1 (from [18]): The mixed Poisson-stable family with real-valued mixing distributions is valid for the stated parameter constraints, giving the PGF in Equation 8.
The sufficiency of the parameter constraints in Theorem 3.1, i.e., that Equation 9 is a proper PGF for α∈(1,2] and for α∈(0,1) with δ≥αγ, is imported from the author's prior preprint [18]. The current paper does not prove this existence result.
standard math Lemma 2.0.1 (Feller): an analytic function is a valid PGF iff G(1)=1, continuity holds, and all derivatives are nonnegative on (0,1).
Used to validate the bSib PGF and to rule out α>2, though the paper's own argument for the latter is flawed.
standard math Stable law characteristic functions and the extreme stable Laplace transform formulas (Nolan; Samorodnitsky-Taqqu).
Used in Lemma 3.0.1 and in the development of the mixed Poisson-stable PGF.
standard math Lemma 2.0.5: a count distribution is discretely infinitely divisible if and only if it is compound Poisson (Feller; Steutel-van Harn).
Used for the compound Poisson representation and for Corollary 4.1.1.
ad hoc to paper In the uniqueness proof, the limit as ρ↑1 of expressions involving δ(ρ) exists and δ(ρ) is constant in ρ.
The proof defines δ=μ((1-ρ^α)^{1/α}-(1-ρ))^{-1}, which depends on ρ, but then treats it as a constant when deriving the differential equation (19). This is an unproved regularity assumption.

invented entities (1)

Broad Sibuya (bSib) distribution
purpose: Provides the summand distribution in the compound Poisson representation of broadly discrete stable distributions.
It is defined by its PGF H(z) and proven to be a valid PGF within the paper. There is no external falsifiable handle; it is a mathematical construction.

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Cite this review

Pith. "Pith review of Broadly discrete stable distributions." pith.science (2026). https://pith.science/paper/RXB5SUZM

@misc{pith2026250905497,
  author       = {Pith},
  title        = {Pith review of: Broadly discrete stable distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXB5SUZM}},
  note         = {Machine review of arXiv:2509.05497}
}

read the original abstract

Stable distributions are of fundamental importance in probability theory, yet their absolute continuity makes them unsuitable for modeling count data. A discrete analog of strict stability has been previously proposed by replacing scaling with binomial thinning, but it only holds for a subset of the tail index parameters. Here, we generalize the discrete stable class to the full range of tail indices and show that it is equivalent to the mixed Poisson-stable family. This broadly discrete stable family is discretely infinitely divisible, with a compound Poisson representation involving a novel generalization of the Sibuya distribution. Under additional parameter constraints, they are also discretely self-decomposable and unimodal. The discrete stable distributions provide a new frontier in probabilistic modeling of both light and heavy tailed count data.

Figures

Figures reproduced from arXiv: 2509.05497 by the authors.

**Figure 1.** Examples of discrete stable probability mass functions. discrete self-decomposable class seems to argue against this approach and we do not advocate it. Alternatively, if we wish to ensure all discrete stable distributions are discretely self-decomposable, perhaps the more restrictive parameter constraints of 4.2 are more appropriate. In addition to exploring the above topics, future work should consider how the di… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 1 Pith paper

[18]

William Townes

F. William Townes. Mixed Poisson families with real-valued mixing distri- butions, July 2024

work page 2024
[1]

Sur les fonctions absolument monotones.Acta Mathemat- ica, 52(none):1–66, January 1929

Serge Bernstein. Sur les fonctions absolument monotones.Acta Mathemat- ica, 52(none):1–66, January 1929

work page 1929
[2]

A triptych of discrete distributions related to the stable law

Luc Devroye. A triptych of discrete distributions related to the stable law. Statistics & Probability Letters, 18(5):349–351, December 1993

work page 1993
[3]

John Wiley & sons., 1968

William Feller.An Introduction to Probability Theory and Its Applications, volume 1. John Wiley & sons., 1968

work page 1968
[4]

John Wiley & Sons, 1971

William Feller.An Introduction to Probability Theory and Its Applications, volume 2. John Wiley & Sons, 1971

work page 1971
[5]

Grandell.Mixed Poisson Processes

J. Grandell.Mixed Poisson Processes. Chapman and Hall/CRC, New York, October 2020

work page 2020
[6]

Gupta and Ed Waymire

Vijay K. Gupta and Ed Waymire. Multiscaling properties of spatial rainfall and river flow distributions.Journal of Geophysical Research: Atmospheres, 95(D3):1999–2009, 1990

work page 1999
[7]

Some Interrelations among Compound and Generalized Distributions.Biometrika, 44(1/2):265–268, 1957

John Gurland. Some Interrelations among Compound and Generalized Distributions.Biometrika, 44(1/2):265–268, 1957

work page 1957

Show all 19 references

[8]

Thierry E. Huillet. On Mittag-Leffler distributions and related stochastic processes.Journal of Computational and Applied Mathematics, 296:181– 211, April 2016

2016
[9]

Kokonendji

Bent Jørgensen and C´ elestin C. Kokonendji. Discrete dispersion models and their Tweedie asymptotics.AStA Advances in Statistical Analysis, 100(1):43–78, January 2016

2016
[10]

Kemp and C

Adrienne W. Kemp and C. D. Kemp. An alternative derivation of the Hermite distribution.Biometrika, 53(3-4):627–628, December 1966

1966
[11]

C. D. KEMP and ADRIENNE W. KEMP. Some properties of the ‘Hermite’ distribution.Biometrika, 52(3-4):381–394, December 1965

1965
[12]

Kozubowski and Krzysztof Podg´ orski

Tomasz J. Kozubowski and Krzysztof Podg´ orski. A generalized Sibuya distribution.Annals of the Institute of Statistical Mathematics, 70(4):855– 887, August 2018

2018
[13]

J. P. Nolan.Stable Distributions - Models for Heavy Tailed Data. Birkhauser, Boston, 2018

2018
[14]

Routledge, 1994

Gennady Samorodnitsky and Murad Taqqu.Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Routledge, 1994

1994
[15]

Generalized hypergeometric, digamma and trigamma dis- tributions.Annals of the Institute of Statistical Mathematics, 31(3):373– 390, December 1979

Masaaki Sibuya. Generalized hypergeometric, digamma and trigamma dis- tributions.Annals of the Institute of Statistical Mathematics, 31(3):373– 390, December 1979. 17 Townes 2025 Broadly discrete stable distributions

1979
[16]

F. W. Steutel and K. van Harn. Discrete Analogues of Self-Decomposability and Stability.The Annals of Probability, 7(5):893–899, 1979

1979
[17]

Steutel and Klaas van Harn.Infinite Divisibility of Probability Distributions on the Real Line

Fred W. Steutel and Klaas van Harn.Infinite Divisibility of Probability Distributions on the Real Line. CRC Press, Boca Raton, October 2003

2003
[19]

Dover Publications, 2010

David Vernon Widder.The Laplace Transform. Dover Publications, 2010. 18 Townes 2025 Broadly discrete stable distributions A Proof of Theorem 3.1 We need to establish equivalence with Definition 3.4. A.1 Proposed distribution has discrete stability property First, assumeXhas a ...

2010

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [18]

William Townes

F. William Townes. Mixed Poisson families with real-valued mixing distri- butions, July 2024

work page 2024

[2] [1]

Sur les fonctions absolument monotones.Acta Mathemat- ica, 52(none):1–66, January 1929

Serge Bernstein. Sur les fonctions absolument monotones.Acta Mathemat- ica, 52(none):1–66, January 1929

work page 1929

[3] [2]

A triptych of discrete distributions related to the stable law

Luc Devroye. A triptych of discrete distributions related to the stable law. Statistics & Probability Letters, 18(5):349–351, December 1993

work page 1993

[4] [3]

John Wiley & sons., 1968

William Feller.An Introduction to Probability Theory and Its Applications, volume 1. John Wiley & sons., 1968

work page 1968

[5] [4]

John Wiley & Sons, 1971

William Feller.An Introduction to Probability Theory and Its Applications, volume 2. John Wiley & Sons, 1971

work page 1971

[6] [5]

Grandell.Mixed Poisson Processes

J. Grandell.Mixed Poisson Processes. Chapman and Hall/CRC, New York, October 2020

work page 2020

[7] [6]

Gupta and Ed Waymire

Vijay K. Gupta and Ed Waymire. Multiscaling properties of spatial rainfall and river flow distributions.Journal of Geophysical Research: Atmospheres, 95(D3):1999–2009, 1990

work page 1999

[8] [7]

Some Interrelations among Compound and Generalized Distributions.Biometrika, 44(1/2):265–268, 1957

John Gurland. Some Interrelations among Compound and Generalized Distributions.Biometrika, 44(1/2):265–268, 1957

work page 1957