On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts

Alexey Khartov

REVIEW 2 major objections 4 minor 26 references

On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that if a rational-infinitely divisible distribution has a non-zero discrete part, then both its own characteristic function and that of the discrete part are bounded away from zero.

desk verdict The paper's central theorem relies on a false almost-periodic lemma, so the main results are not established as written; the surrounding lemmas and Proposition 2 are solid. read the letter →

arxiv 2509.07244 v1 pith:JJZD5FUZ submitted 2025-09-08 math.PR

Alexey Khartov This is my paper · ORCID

classification math.PR MSC 60E1060E0742A75

keywords rational-infinitedivisibilityquasi-infinitecharacteristicfunctionsLévy–Khintchine-typerepresentationsignedspectralmeasurediscretedistributionsalmostperiodicseparationfromzero

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 studies rational-infinite divisibility: probability laws that are convolution factors of infinitely divisible laws, so their characteristic functions are quotients of infinitely divisible characteristic functions. It proves that whenever such a law has a non-zero discrete part, the full characteristic function cannot come arbitrarily close to zero, and neither can the characteristic function of the discrete part. In symbols, for $F\in \mathcal{Q}$ with $c_d>0$, the paper establishes $\inf_{t\in\mathbb{R}} |f(t)|>0$ and $\inf_{t\in\mathbb{R}} |f_d(t)|>0$. These are necessary conditions, offered as steps toward a complete membership criterion for the class $\mathcal{Q}$ when a discrete part is present.

What carries the argument

The mechanism is a comparison of two different averages of characteristic-function values. The signed spectral representation (1) yields Lemma 1: the quotient $|f(t-h)f(t+h)|/|f(t)|^2$ is bounded by $Ce^{Bh^2}$ with constants independent of $t$, because the signed spectral function $G$ has bounded total variation. Lemma 2 shows that for any continuous distribution, the Cesàro averages $(2\tau)^{-1}\int_{-\tau}^{\tau}|f_c(t+h)|^2\,dh$ vanish uniformly in $t$ as $\tau\to\infty$, by writing $|f_c|^2$ as the characteristic function of the symmetrized law and applying dominated convergence. The discrete part contributes the opposite effect: $f_d$ is almost periodic, so its translates have uniformly convergent subsequences, and the limiting function $\varphi$ is not identically zero, giving a positive mean-square constant $A$ in (11). The two main theorems force these opposite behaviours into contradiction unless both $f$ and $f_d$ stay away from zero.

What would settle it

The theorems can be settled by one explicit calculation. Take $f(t)=c_d(1+e^{it})/2+(1-c_d)e^{-|t|}$ with $0<c_d<1$: this is a characteristic function of a two-point discrete part mixed with a Cauchy law, it has no zeros, yet $\inf_t |f(t)|=\inf_t |f_d(t)|=0$. Determine whether $f$ equals a quotient $f_1/f_2$ of infinitely divisible characteristic functions, equivalently whether $f$ admits the signed-spectral representation (1) with a bounded-variation function $G$. The theorems assert that it does not; exhibiting the representation would disprove them, and proving impossibility would confirm the necessary conditions.

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Extended reading notes

Core claim

The central claim is Theorem 1 and Theorem 2. Let $F=c_dF_d+(1-c_d)F_c$ be the Lebesgue decomposition of a rational-infinitely divisible law, with $c_d>0$, and let $f,f_d,f_c$ be the corresponding characteristic functions. The paper proves that $\inf_{t\in\mathbb{R}} |f(t)|>0$ and, as a consequence, $\inf_{t\in\mathbb{R}} |f_d(t)|>0$. The proof of Theorem 1 argues by contradiction: a sequence $t_k\to\infty$ with $|f(t_k)|\to 0$ would make the averaged quotient in (5) unbounded, contradicting the uniform bound supplied by the signed-spectral representation together with the positive mean-square energy of the almost periodic limit of translates of $f_d$. Theorem 2 follows by combining Theorem 1 with a structural fact proved for arbitrary distributions: whenever the full characteristic function is separated from zero, the discrete-part characteristic function is separated from zero too.

Load-bearing premise

The proof of Theorem 1 relies on an imported fact from earlier work, not re-proven here: a uniformly convergent subsequence of translates of the discrete characteristic function cannot converge to the identically zero function, so the mean-square constant $A$ is strictly positive; if that fact failed, the lower bound on the discrete contribution would collapse and the contradiction argument would not go through.

Editorial extensions

If this is right

If $F\in\mathcal{Q}$ and $c_d>0$, then the full characteristic function satisfies $\inf_{t}|f(t)|>0$; even a slow approach to zero at infinity is impossible.
Under the same assumptions the discrete part also satisfies $\inf_{t}|f_d(t)|>0$, so by the earlier discrete criterion the discrete component is itself rational-infinitely divisible.
Within the class $\mathcal{Q}$, the presence of a discrete part is equivalent to separation of $f$ from zero: $\inf_{t}|f(t)|>0$ exactly when $c_d>0$ (Corollary 1).
Any future sufficient condition for membership in $\mathcal{Q}$ with $c_d>0$ must include $\inf_{t}|f(t)|>0$; once that holds, $\inf_{t}|f_d(t)|>0$ follows automatically from Proposition 2.

Reading between the lines

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

One can test the sharpness of the theorems on the mixture $f(t)=c_d(1+e^{it})/2+(1-c_d)e^{-|t|}$: it is a zero-free characteristic function with a two-point discrete part and $\inf|f|=\inf|f_d|=0$, so the theorems predict it cannot admit a signed spectral representation; confirming the failure would show that zero-free is far from sufficient once an atom is present.
The same averaging strategy likely transfers to multivariate quasi-infinitely divisible laws and to locally compact abelian groups, where almost periodic functions and mean values have standard analogues; the necessary separation conditions should take the same form.
A quantitative refinement would estimate how large $\inf_t|f(t)|$ must be from the total variation of the signed spectral measure $G$; the constants in Lemma 1 provide a starting point for such a bound.

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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

The short version: this paper does not, as written, prove what it claims. The intended results are two necessary conditions for rational-infinite divisibility when the distribution has a nonzero discrete part. These are plausible and would be a genuine incremental advance over earlier work by Berger–Kutlu and by Khartov himself, which needed extra assumptions on the continuous part. The packaging is clean: Lemma 2, the uniform Cesàro decay for continuous characteristic functions, is proved correctly, and Proposition 2 — inf |f|>0 implies inf |f_d|>0 — has a solid, self-contained argument using almost-periodicity and the continuity of the symmetrized continuous part. If Theorem 1 were available, Theorem 2 would follow immediately.

The problem is Theorem 1. The proof assumes inf |f|=0, picks t_k with |f(t_k)|→0, passes to a uniform limit φ of translates of f_d, and then invokes a fact from the author's earlier paper [11]: the product φ(h)φ(−h) is not identically zero. That fact is false. Take the periodized triangular function G(t)=Σ_n max(0,1−|t−2πn|). This is a perfectly good lattice characteristic function: its Fourier coefficients are nonnegative and sum to 1. Shift by t_k=π/2+2πk. Every translate is exactly G(h+π/2), so φ(h)=G(h+π/2). The supports of φ(h) and φ(−h) are disjoint intervals around −π/2 and +π/2 modulo 2π, so φ(h)φ(−h)≡0. The constant A in the proof is zero, the lower bound on J_d collapses, and the contradiction I(t_k,τ)→∞ never gets off the ground. The proof does not use anything about F∈Q to rule out this situation; it simply imports a false statement.

This is a central gap, not a cosmetic one. Theorem 2 depends on Theorem 1, so the paper's main contribution is currently unsupported. I want to be fair: the rest of the machinery — Lemma 1, Lemma 2, Proposition 2, Corollary 1 — looks fine, and the author has engaged honestly with the prior literature. The reader report gave soundness 8; I think that over-scores the paper once the stress-test counterexample is checked.

Who should read this? Specialists in quasi-infinitely divisible distributions, mainly as a caution about relying on the imported almost-periodic lemma. If the author can replace that lemma with a correct argument, the intended theorem would be worth publishing. As it stands, I would not accept it, and I would not cite it. For peer review: I would send it to a referee who knows almost periodic functions — they would quickly flag the bad lemma and the author might repair the proof — but the current version should not be published.

Referee Report

2 major / 4 minor

Summary. The paper studies the class Q of rationally (or quasi-) infinitely divisible distributions. Its main results are two necessary conditions: Theorem 1 states that if F ∈ Q and its discrete part has positive mass c_d > 0, then inf_{t∈R} |f(t)| > 0; Theorem 2 states that in the same situation inf_{t∈R} |f_d(t)| > 0. The proof of Theorem 1 combines an upper Cesàro bound derived from the Lévy–Khintchine-type representation (Lemma 1) with a lower bound involving the discrete and continuous parts. The discrete contribution is analyzed by passing to a uniformly convergent subsequence of translates of f_d, and the continuous contribution is controlled by a uniform Cesàro vanishing result (Lemma 2). Proposition 2, which derives inf |f_d| > 0 from inf |f| > 0, is proved separately by an almost-periodicity and mean-value argument, and Theorem 2 is then obtained as a direct consequence of Theorem 1 and Proposition 2.

Significance. If the two theorems are correct, they are natural and potentially useful necessary conditions for rational-infinite divisibility in the presence of a discrete part, and they would complement the known sufficient conditions in this area. The paper is clearly written, Lemma 2 is proved correctly by a neat symmetrization argument, and Proposition 2's proof appears sound and self-contained. The main weakness is that the proof of Theorem 1 relies on an imported almost-periodicity assertion that is false as stated; since Theorem 2 depends on Theorem 1, the central claims are not established by the submitted argument. The paper may be worth publishing after a substantial repair of this step, but in its current form it is not acceptable.

major comments (2)

[Theorem 1, Proof, Equations (9)–(12)] The assertion after (9) that a uniform limit φ of translates of f_d must satisfy φ(h_0)φ(−h_0) ≠ 0 for some h_0 is false. Let G(t) = Σ_{n∈Z} max(0, 1 − |t − 2πn|). This is the characteristic function of a discrete law: its Fourier coefficients are p_k = (1/2π)(sin(k/2)/(k/2))², which are nonnegative and sum to 1. Choose t_k = π/2 + 2πk. Then φ_k(h) = G(t_k + h) = G(h + π/2) for every k, so the uniform limit is φ(h) = G(h + π/2). The positivity set of φ(h) is 2πZ + (−π/2 − 1, −π/2 + 1), while the positivity set of φ(−h) is 2πZ + (π/2 − 1, π/2 + 1); these sets are disjoint. Hence φ(h)φ(−h) ≡ 0, so the constant A in (11) is zero, the lower bound in (12) gives no positive mass, and the claimed contradiction I(t_{k_l}, τ) → ∞ is not obtained. The proof does not use any additional property of f_d coming from F ∈ Q at this point, so the citation to [11] cannot justify the step; the authors need either to prove a correct version under the actual hypotheses or to replace the argument.
[Theorem 2] Theorem 2 is derived as a direct consequence of Theorem 1 and Proposition 2. Since the proof of Theorem 1 has the gap described above, the proof of Theorem 2 is incomplete as it stands. Proposition 2's own proof does not depend on the problematic almost-periodicity step and appears correct; the missing work is in Theorem 1.

minor comments (4)

[Introduction, around (2)] The phrase 'Here the coefficients c_a, c_a, and c_s' should read c_d, c_a, and c_s; the coefficient c_d is missing.
[Proof of Proposition 1] The symbol μ_d is introduced for inf_{t∈R} |f(t)|, but μ_d was already used in the Introduction for inf_{t∈R} |f_d(t)|. Please rename one of them to avoid confusion.
[Throughout] There are several typos and grammatical slips: 'separeted' in the Introduction, 'simmetrization' in Lemma 2, 'arbitarary' in the proof of Theorem 1, and 'it have already found' in the Abstract.
[Lemma 2] The opening phrase 'It is always valid that' is awkward; the statement is a uniform Cesàro-vanishing property of |f_c|² and should be phrased accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proven from the Lévy–Khintchine-type representation and self-contained lemmas, and the auxiliary self-citations are independent mathematical facts rather than re-statements of the target results.

full rationale

The paper's central claims are Theorem 1 (F∈Q and c_d>0 implies inf |f|>0) and Theorem 2 (F∈Q and c_d>0 implies inf |f_d|>0). The proof of Theorem 1 is not circular: Lemma 1 is derived in the paper from the signed Lévy–Khintchine representation, and Lemma 2 is proven by writing |f_c|^2 as the characteristic function of a continuous symmetrized law and applying dominated convergence. The contradiction argument uses the decomposition f = c_d f_d + (1−c_d)f_c, the almost-periodicity of f_d, and Lemma 2. The only imported results are auxiliary facts about almost periodic functions cited to the author's earlier work [11] and to Levitan [18]: relative compactness of translates, existence of h0 with φ(h0)φ(−h0)≠0, and Parseval identity for almost periodic functions. These facts do not contain the target conclusions inf |f|>0 or inf |f_d|>0 and are not fitted to the data of the theorem. Proposition 1 and Proposition 2 are proven directly. Therefore no step reduces by construction to its own input, and no fitted parameter is renamed as a prediction. A concern about the correctness of the imported almost-periodic lemma would be a proof gap, not circularity; the circularity score remains 0.

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

The paper introduces no free parameters or invented entities. All auxiliary inputs are standard theorems from probability and almost periodic function theory; two of them are cited to the author's own prior work [11] and [12], but they are technical lemmas, not the paper's target results.

assumptions (5)

domain assumption Lévy-Khintchine-type representation (1) for F ∈ Q with a signed spectral function G of bounded total variation, and uniqueness of the spectral pair (γ, G).
Used at the start of Lemma 1 and throughout; sourced from [10] and [19], not re-derived.
domain assumption The characteristic function f_d of a discrete distribution is almost periodic, and certain structural properties of its uniformly convergent translate limits, such as the existence of h with φ(h)φ(−h) ≠ 0 and the Parseval identity (11) with A > 0.
Used in Theorem 1 and Proposition 2; cited to [11] and [18], not proved in this note.
standard math The inequality (1 − cos(hx))(1 + x^2)/x^2 ≤ h^2/2 + 2 for all real x and h.
Used in Lemma 1 to produce the bound with B and C; cited to [12].
standard math Bohr's theorem on relatively dense sets of ε-almost periods for almost periodic functions.
Used in Proposition 2 to construct the sequence t_{ε,k} with control on |f_d|; cited to [18].
standard math Lebesgue's dominated convergence theorem and Fubini's theorem.
Used in Lemma 2 and in interchanging integrals.

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Pith. "Pith review of On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts." pith.science (2026). https://pith.science/paper/JJZD5FUZ

@misc{pith2026250907244,
  author       = {Pith},
  title        = {Pith review of: On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JJZD5FUZ}},
  note         = {Machine review of arXiv:2509.07244}
}

abstract

We consider the new class $\boldsymbol{Q}$ of rational-infinitely (or quasi-infinitely) divisible distribution functions on the real line. By definition, $F\in \boldsymbol{Q}$ if there are some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$, where ``$*$'' is the convolution. The characteristic function of such $F$ admits the L\'evy--Khintchine-type representation with a ``signed spectral measure''. The class $\boldsymbol{Q}$ is a significant extension of the family of infinitely divisible distribution functions and it have already found some applications in several areas. So there is an active interest in this class. In particular, a lot of results have recently appeared on the problem of belonging to the class $\boldsymbol{Q}$ in terms of characteristic functions. In the paper, we continue this series of results by proposing two necessary conditions for distribution functions from $\boldsymbol{Q}$ with non-zero discrete parts. Namely, the characteristic functions of such a distribution function and its discrete part are always separated from zero.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [11]

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

work page 2022

[2] [1]

I. A. Alexeev, A. A. Khartov,Spectral representations of characteristic functions of discrete probability laws, Bernoulli29(2023), 2, 1392–1409

work page 2023

[3] [2]

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

work page 2023

[4] [3]

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

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

work page 2019

[5] [4]

Berger, M

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

work page 2023

[6] [5]

Berger, M

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

work page 2021

[7] [6]

Invertible Complex Measures on Euclidean Spaces

D. Berger, A. Lindner,Invertible complex measures on Euclidean spaces, (2025), arXiv:2506.09012

work page Pith review arXiv 2025

[8] [7]

Chhaiba, N

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

work page 2016