{"id":"48248edd-7a3a-440e-aa5f-7b948f66e970","arxiv_id":"2509.05497","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Discrete stability is generalized to all tail indices and shown to be equivalent to the mixed Poisson-stable family, with a compound Poisson representation.","lead":"This paper extends discrete stable distributions, previously limited to tail indices at most 1, to the full range 0 to 2 by adding Poisson translation to the stability operation, and proves these are exactly the mixed Poisson-stable family. It gives these distributions a compound Poisson form with a new summand distribution and identifies when they are discrete self-decomposable.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniqueness proof in Appendix A.2.2 silently treats the ρ-dependent shift δ as constant when taking the limit that yields Eq. (19); without proving δ_ρ is constant or convergent, the characterization theorem is not established.","rationale":"I read the paper in good faith: the forward direction (Appendix A.1) is a direct verification and is solid; the compound Poisson representation (Theorem 4.1) gives an independent existence proof for the whole family, so the existence side is not the main risk. The central claim is the uniqueness in Theorem 3.1. The reader's identification of the ρ-dependent δ in Appendix A.2.2 is correct and is the most load-bearing issue: the limit step that produces the differential equation silently assumes a constant δ. Without a proof that δ_ρ is constant or has a controlled limit, the 'only if' direction is not established. I do not treat the 'Var=E[X] implies Poisson' assertion as equally load-bearing: it is false in general, but the exclusion of α>2 could likely be obtained from the sign/analyticity of the PGF coefficients, so it is a repairable secondary flaw. The paper explicitly relies on a self-cited preprint [18] for the mixed Poisson-stable equivalence, but Theorem 4.1 provides an internal construction; still, the uniqueness proof gap is enough to keep the paper at CONDITIONAL rather than ACCEPT. My recommendation is UNCHANGED relative to the reader's verdict.","tokens_in":16023,"tokens_out":15170,"duration_ms":152580,"concrete_test":"Re-derive the limit in Appendix A.2.2 with δ explicitly indexed by ρ. Define u_ρ=(1-ρ^α)^{1/α}, h=(1-ρ)(1-z), and compute lim_{ρ↑1} [G(z+h)e^{δ_ρ(1-z-h)}-G(z)e^{δ_ρ(1-z)}]/[hG(z+h)e^{δ_ρ(1-z-h)}] keeping δ_ρ in the expression, under only the assumptions that G is a PGF and Definition 3.4 holds. If this limit can be shown to equal G'(z)/G(z)-δ* with δ* a constant independent of ρ, the proof gap is closed. If it does not, or if the limit depends on the path of ρ, then Eq. (19) is not a consequence of the discrete-stability equation and the uniqueness claim needs an additional argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The uniqueness direction of Theorem 3.1 (Appendix A.2.2) defines δ via the translation parameter μ_ρ by δ_ρ = μ_ρ ((1-ρ^α)^{1/α}-(1-ρ))^{-1}, so δ carries a ρ-subscript. It then rearranges the discrete-stability equation into G(z)e^{δ_ρ(1-z)} = G(1-ρ(1-z))e^{ρδ_ρ(1-z)} G(1-u_ρ(1-z))e^{u_ρδ_ρ(1-z)} with u_ρ=(1-ρ^α)^{1/α}. Passing to the limit ρ↑1, the proof sets h=(1-ρ)(1-z) and writes the right-hand side as [G(z+h)f(z+h)-G(z)f(z)]/[hG(z+h)f(z+h)], f(z)=e^{δ(1-z)}, whose limit is G'(z)/G(z)-δ. This computation is valid only if δ_ρ is independent of ρ, or at least if the ρ-dependence is controlled so that a single limiting δ appears. The manuscript proves neither. Consequently the differential equation (19), and the PGF form obtained by integrating it, are not derived from Definition 3.4. This is the load-bearing gap in the claimed characterization. (The separate assertion in the proof that Var[X]=E[X] implies Poisson is also false; it is used to exclude α>2, though a direct coefficient argument seems available.)","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16455,"tokens_out":13040,"duration_ms":134232,"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":[{"comment":"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.","section":"Appendix A.2.2 and A.2.3"},{"comment":"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.","section":"Appendix A.2.2, exclusion of α>2"},{"comment":"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].","section":"Theorem 3.1 and Proposition 3.1"}],"minor_comments":[{"comment":"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.","section":"Section 4.2"},{"comment":"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.","section":"Appendix A.2.2"},{"comment":"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).","section":"Theorem 3.1"},{"comment":"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.","section":"References"},{"comment":"The six panels in Figure 1 are small and the parameter labels are cramped; larger panels or separate rows would improve readability.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the ρ-dependence of δ in the uniqueness proof. If the author can show that δ_ρ is actually constant (e.g., by evaluating the defining equation at z=0 and using G'(0)>0 for non-Poisson solutions), or supply a different uniqueness argument, the central claim becomes plausible. The false Poisson implication is easily repaired by a direct coefficient argument. The reliance on [18] for the sufficiency of the PGF constraints should be checked by the editor; the paper would be stronger if that part were made self-contained or clearly delimited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nSoft 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.\n\nWho 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.","headline":"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.","tokens_in":16843,"tokens_out":5015,"would_cite":false,"duration_ms":46703,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E07","62E10","60E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete stable laws now cover all tail indices from 0 to 2","keywords":["discrete stable distributions","mixed Poisson-stable family","binomial thinning","Poisson translation","discrete infinite divisibility","Sibuya distribution","self-decomposability","count data"],"falsifier":"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.","tokens_in":1773,"feed_emoji":"🎲","tokens_out":4299,"duration_ms":118144,"temperature":0.7,"pith_summary":"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.","feed_headline":"Discrete stable laws now cover all tail indices from 0 to 2","feed_subtitle":"They are exactly mixed Poisson-stable, one model for both light- and heavy-tailed counts.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines binomial-thinning stability and the strict discrete stable family that this paper generalizes.","marker":"[16]"},{"why":"Establishes mixed Poisson families with real-valued mixing distributions and gives the Poisson-stable PGF that is identified as the unique discrete stable family.","marker":"[18]"},{"why":"Introduces the dilation and Poisson translation operations used in the definition of broad discrete stability.","marker":"[9]"},{"why":"Provides the stable distribution parameterization and tail properties underlying the extreme stable mixing distribution.","marker":"[13]"},{"why":"Gives the bilateral Laplace transform form of extreme stable laws used in the mixed Poisson representation.","marker":"[14]"},{"why":"Provides the R-function framework and discrete infinite divisibility / self-decomposability concepts used throughout.","marker":"[17]"},{"why":"Supplies the equivalence between discrete infinite divisibility and compound Poisson representation used in the paper's corollary.","marker":"[3]"},{"why":"Defines the Sibuya distribution whose broad generalization is the compound Poisson summand.","marker":"[15]"}],"fun_headline_variants":["Discrete stable laws span every tail index","Discrete stable equals mixed Poisson-stable for all tails","New broad discrete stable class covers index 0 to 2","Stable count models now full-range: light to heavy tails"],"cache_read_input_tokens":18560,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Discrete stable laws span every tail index","Discrete stable equals mixed Poisson-stable for all tails","New broad discrete stable class covers index 0 to 2","Stable count models now full-range: light to heavy tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000357,"raw_usage":{"total_tokens":1745,"prompt_tokens":687,"completion_tokens":1058,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":992}},"tokens_in":431,"tokens_out":1058,"duration_ms":10601,"temperature":1.0,"reasoning_tokens":992,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T05:26:10.613235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines binomial-thinning stability and the strict discrete stable family that this paper generalizes."},{"cited_title":"William Townes","cited_arxiv_id":null,"evidence_quote":"Establishes mixed Poisson families with real-valued mixing distributions and gives the Poisson-stable PGF that is identified as the unique discrete stable family."},{"cited_title":"Kokonendji","cited_arxiv_id":null,"evidence_quote":"Introduces the dilation and Poisson translation operations used in the definition of broad discrete stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the stable distribution parameterization and tail properties underlying the extreme stable mixing distribution."},{"cited_title":"Routledge, 1994","cited_arxiv_id":null,"evidence_quote":"Gives the bilateral Laplace transform form of extreme stable laws used in the mixed Poisson representation."},{"cited_title":"Steutel and Klaas van Harn.Infinite Divisibility of Probability Distributions on the Real Line","cited_arxiv_id":null,"evidence_quote":"Provides the R-function framework and discrete infinite divisibility / self-decomposability concepts used throughout."},{"cited_title":"John Wiley & sons., 1968","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between discrete infinite divisibility and compound Poisson representation used in the paper's corollary."},{"cited_title":"Generalized hypergeometric, digamma and trigamma dis- tributions.Annals of the Institute of Statistical Mathematics, 31(3):373– 390, December 1979","cited_arxiv_id":null,"evidence_quote":"Defines the Sibuya distribution whose broad generalization is the compound Poisson summand."}],"review_version":1}