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Fractional counting process at L\'evy times and its applications

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Pith's one-line read The paper constructs the time-changed fractional counting process by subordinating the recently introduced fractional counting process with an independent Lévy subordinator, and derives explicit formulas for its probability mass function…

desk verdict Standard subordination results with a missing moment condition; the paper's own stable-subordinator example breaks its central formulas. read the letter →

arxiv 2412.04334 v1 pith:LQTJH3O4 submitted 2024-12-05 math.PR

Shilpa Garg , Ashok Kumar Pathak , Aditya Maheshwari This is my paper · ORCID
classification math.PR MSC 60G2260G5511B7360K10
keywords fractionalcountingprocessLévysubordinatortime-changedMittag-LefflerfunctionsubordinatedBellpolynomialswaitingtimedistributionfirstpassageshockdeteriorationmodel
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 introduces the time-changed fractional counting process (TCFCP), obtained by running the recently introduced fractional counting process (FCP) on a random clock given by an independent Lévy subordinator. It derives an explicit probability mass function for the TCFCP as a double series in fractional moments of the subordinator, along with its Laplace transform, probability generating function, moment generating function, mean, and variance. Waiting-time and first-passage-time distributions are computed, and multiplicative and additive compound versions of both the FCP and the TCFCP are analyzed. The paper also defines subordinated generalized fractional Bell polynomials and shows that the moments of the TCFCP are exactly these polynomials. A shock deterioration model for aging structures is proposed, in which the FCP supplies the arrival process and gamma-distributed increments model the damage per shock.

What carries the argument

The engine of the argument is the probability mass function of the base FCP, built from the generalized three-parameter Mittag-Leffler function $E^{\zeta}_{\mu,\vartheta}(z)$. Subordinating that pmf by a Lévy subordinator replaces every power $y^{\theta(n+k)}$ with the fractional moment $\mathbb{E}[(H(t))^{\theta(n+k)}]$, and the same substitution is applied to the generating functions, so every distributional identity for the base process lifts directly to the time-changed process. The subordinated generalized fractional Bell polynomials are defined from the same series and serve as the moment representation of the TCFCP.

What would settle it

Choose an $\alpha$-stable subordinator with $\alpha=1/2$ and parameters $\theta=1$. For this subordinator $\mathbb{E}[H(t)^p]$ is finite only for $p<\alpha$, so the terms of the pmf series in Eq. (10) with $n+k\ge1$ are undefined. A concrete check: compute the truncated series for $z(1,t)$ and see whether it converges, or verify whether $\sum_{n=0}^N z(n,t)$ tends to 1 as $N\to\infty$; if the moments are infinite, the series cannot define a probability mass function. The paper's Remark 5.1 applies the stable moment formula for all orders, so evaluating that formula for $n+k$ above $\alpha$ should produce a contradiction.

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Extended reading notes

Core claim

The central claim is that subordinating the FCP by an independent Lévy subordinator $H(t)$ produces a process $Z(t)=N^{\zeta,\theta}_{\mu,\vartheta}(H(t))$ whose law is completely described by the series $z(n,t)=\frac{(\zeta)_n\Gamma(\vartheta)\lambda_\theta^n}{n!}\sum_{k=0}^\infty\frac{(-\lambda_\theta)^k(\zeta+n)_k}{k!\Gamma(\mu(n+k)+\vartheta)}\,\mathbb{E}[(H(t))^{\theta(n+k)}]$, with the same fractional moments $\mathbb{E}[(H(t))^{m\theta}]$ entering the Laplace transform, pgf, mgf, mean, and variance. The paper further claims that integer moments of the TCFCP are represented by the subordinated generalized fractional Bell polynomials, and that the multiplicative and additive compound variants inherit the same series form after replacing the subordinator moments. These formulas are intended as a unified description of fractional counting processes under Lévy subordination.

Load-bearing premise

Every formula in the paper assumes the Lévy subordinator has finite fractional moments $\mathbb{E}[H(t)^{\theta(n+k)}]$ for all $n,k$ appearing in the infinite series, and that the series and integrals can be interchanged; the paper never states this, and for stable subordinators the usual moment formula only holds below the stability index.

Editorial extensions

If this is right

  • For any Lévy subordinator with the required fractional moments, the full distribution of the TCFCP is available in closed form, so fitting or simulating the process reduces to estimating the subordinator's moments.
  • The mean and variance formulas give an explicit overdispersion structure, allowing the four FCP parameters plus the subordinator to match a wide range of count-data behavior.
  • Waiting-time and first-passage-time distributions follow from the same series, giving direct tools for reliability, ruin, and shock-model calculations.
  • When the subordinator is an $\alpha$-stable process with $\alpha=1$ and $t^\theta$ is replaced by $x$, the subordinated Bell polynomials reduce to the ordinary generalized fractional Bell polynomials, connecting the new process to the prior combinatorial framework.
  • The shock deterioration model with gamma-distributed increments extends the compound Poisson shock model; setting $\mu=\vartheta=\zeta=\theta=1$ recovers that model exactly.

Reading between the lines

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

  • The derivation assumes all fractional moments of the subordinator are finite; for $\alpha$-stable subordinators the standard moment formula is only valid for orders below $\alpha$, so the pmf series may fail to define a probability distribution unless the parameter range is restricted. This is an editorial caveat, not a claim of the paper.
  • A direct numerical check for a gamma subordinator (all moments finite) would validate the pmf formula by comparing the series with Monte Carlo simulation of $N(H(t))$; no such simulation is reported in the paper.
  • The same substitution mechanics could be applied to other count processes, such as negative binomial or generalized fractional variants, to produce time-changed families with explicit moments.
  • The shock model could be extended to the time-changed FGCP at Lévy times; the paper stops at the FCP-driven shock model.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 6 minor

Summary. The paper studies Laskin's generalized fractional counting process (FCP), derives additional properties of it, and then introduces a time-changed fractional counting process (TCFCP) Z(t)=N(H(t)) obtained by subordinating the FCP by an independent Lévy subordinator H. The central results are the claimed explicit probability mass function (Eq. 10), Laplace transform (Eq. 12), probability generating function, mean and variance (Theorem 4.1), as well as waiting-time and first-passage-time formulas, compound variants at Lévy times, subordinated generalized fractional Bell polynomials (SGFBP), and a shock deterioration model. The paper's main contribution is the set of distributional formulas for the TCFCP expressed in terms of fractional moments of the subordinator, together with the SGFBP connection and the application section.

Significance. If the central formulas were valid under a clearly stated set of assumptions, the paper would provide a useful unifying framework for fractional counting processes under Lévy subordination. The basic conditioning idea is natural, and the mean and variance expressions follow from standard conditional expectation once the needed moments exist. The introduction of SGFBP and the shock deterioration model with a bounded series function are interesting extensions. However, as written, the central formulas require unstated finiteness of all fractional moments of the subordinator, and a main example (the α-stable subordinator) violates that condition. In addition, several proofs condition on discrete values of a continuous subordinator and contain ill-posed definitions. These are load-bearing mathematical issues rather than presentation problems.

major comments (6)
  1. [§4.1, Eqs. (10) and (12)] The derivation of the pmf and Laplace transform interchanges an infinite series with the expectation over H(t), producing terms E[(H(t))^{θ(n+k)}] and E[(H(t))^{mθ}]. No condition ensuring finiteness of these fractional moments is stated. For the α-stable subordinator S_α(t) used later in Remark 5.1, E[S_α(t)^p] is finite only for p<α, so for every fixed n and all sufficiently large k the expectation in Eq. (10) is infinite, and the same happens in Eq. (12) for large m. The integral representation z(n,t)=∫ P_{μ,ϑ}^{ζ,θ}(n,y) h(y,t)dy may still define a finite probability, but it is not equal to the displayed series. The paper must either impose an explicit all-fractional-moments assumption (for example, a gamma subordinator) throughout Section 4 and adjust the examples accordingly, or provide a regularized/truncated statement for subordinators with only finitely many fractional moments.
  2. [§4.2, Proposition 4.3 and §4.3, Proposition 4.4] Both proofs condition on P[H(t)=n] and sum over n as though the Lévy subordinator H(t) were a discrete random variable. For the stable, gamma, and tempered stable subordinators considered in the paper, H(t) is absolutely continuous, so P[H(t)=n]=0 and the sums over n of P[H(t)=n] are not meaningful. The final expressions involving E[(H(t))^{θ(...)}] are formally what one would obtain by integrating against the density h(y,t)dy and interchanging sums, but the derivations as written are invalid and need to be rewritten with integrals and explicit justification of the interchange.
  3. [§4.3, Theorem 4.3] The first passage time is defined as T_w := inf{t ≥ 0 : z(n,t) ≥ w}, but z(n,t) is the deterministic probability mass function of Z(t), not the process Z(t) itself. The definition should be T_w := inf{t ≥ 0 : Z(t) ≥ w}. The subsequent computation P[T_w > t] = Σ_{n=0}^{w-1} z(n,t) is the correct survival probability for that process definition, so the final formula is salvageable, but the statement and proof currently use the undefined expression P[z(n,t) < w] and sum P[z(n,t)] over n.
  4. [§5.3, moments of TCFCP] The displayed equality E{[Z(t,λ_θ)]^p} = B_{SG}(λ_θ(H(t))^θ,m) equates a deterministic moment on the left with a random variable on the right. What is actually true, and consistent with the definition of the SGFBP, is the conditional moment E[Z(t)^p | H(t)] = B_{SG}(λ_θ(H(t))^θ,m), or alternatively the unconditional moment E[Z(t)^p] = E[B_{SG}(λ_θ(H(t))^θ,m)]. In addition, Eq. (11) defines z(n,x) with no x appearing on the right-hand side; the notation must be fixed so that the argument of B_{SG} is well-defined.
  5. [Remark 5.1] The formula E[S_α(t)^p] = Γ(1−p/α)/Γ(1−p) t^{p/α} is cited for p>0, but for an α-stable subordinator this identity is valid only for 0<p<α; for p≥α the moment is infinite. Since the SGFBP series in Eq. (23) includes powers θ(n+k) for arbitrarily large n+k, the α-stable case is not a valid example for the displayed series. The same problem affects the incomplete gamma subordinator in Remark 5.1(ii), where the asymptotic is cited only for p≤α. These remarks should be corrected or replaced with subordinators that have finite fractional moments of all orders.
  6. [Remark 4.2] The proof that the TCFCP does not have independent increments uses the equality E[(H(t_1+t_2))^{mθ}] = E[(H(t_1))^{mθ}] E[(H(t_2))^{mθ}]. This is not a valid identity for Lévy subordinators: independent increments factor the Laplace transform of the sum, not the moments of the sum. The conclusion may be true, but the displayed argument is incorrect and needs to be replaced by a direct computation involving the joint Laplace transform of the increments.
minor comments (6)
  1. [Title] The title contains a typo: 'A t L ´evy times' should read 'At Lévy times'.
  2. [References] Several references contain corrupted author-name artifacts such as 'Wy/suppress loma´ nska' (refs. [15], [21]); these should be cleaned up.
  3. [§4.1, Proposition 4.2] The function H(s,t) is called the mgf but is defined with e^{-sn}; either use e^{sn} or call it the Laplace transform / generating function, and align the notation with Eq. (13).
  4. [§4.2, Example 4.2] In the beta-product example, after taking c=d_1=...=d_m=1, the density f_{R_m}(x) is written with B(1,m) in the denominator; please check the normalization and whether it should be B(1,m) or another expression.
  5. [§4.2, Corollary 4.1(i)] The density h_{Z_π}(y,t) in Eq. (17) has support restrictions (y≠0 or y≠1 depending on the atom at 1) that are not fully spelled out; please state the domain of the density explicitly.
  6. [§4.3, Corollary 4.4] In Example 4.4, the Poisson convolution formula b_s^{*m} = e^{-mρ}(mρ)^s/s! is stated for s ∈ N_0; it would be clearer to indicate that this is the standard compound Poisson convolution.

Circularity Check

1 steps flagged · score 2.0 of 10

One definitional moment identity; core TCFCP derivation is self-contained.

  1. self definitional [Section 5.3, Moments of TCFCP (following Eq. (23))]
    "By the definition of the pth order moment, we have that E{[Z(t,λθ)]^p} = Σ_{n=0}^∞ n^p z(n,t). It can be easily followed that E{[Z(t,λθ)]^p} = BSG(λθ(H(t))^θ,m)."

    Eq. (23) in Section 5.1 defines BSG(x,m) as Σ_n n^m z(n,x), where z(n,x) is the pmf of the time-changed FCP for a generic random variable X (Eq. (11)). Therefore the Section 5.3 assertion that TCFCP moments equal BSG(...) is exactly the same sum written with z(n,t); the equality holds by the definition of BSG, not by an independent structural property. This is a definitional equivalence, and it does not affect the Section 4 derivations of Eq. (10), Eq. (12), or Theorem 4.1, which are obtained by conditioning on H(t).

full rationale

The central Section 4 derivation of the TCFCP pmf, Laplace transform, pgf, mgf, mean, and variance is obtained by conditioning on the independent Lévy subordinator H(t) and substituting Laskin's FCP pmf/pgf; these are standard subordination computations and do not presuppose the target formulas. The only place where the paper's equations reduce to their own definitions is the Bell-polynomial connection: Eq. (23) defines BSG(x,m) as the m-th moment sum of the TCFCP pmf z(n,x), so Section 5.3's statement E[Z^p] = BSG(...) is true by construction. This is a definitional renaming of the moment formula, not an externally derived relationship, and it is not load-bearing for the main results or the shock-model application. Self-citations (e.g., [32], [40]) appear but are not used as the sole justification of any theorem. Consequently, no prediction reduces by construction and no load-bearing self-citation chain exists. The α-stable moment issue raised in the skeptic note is a correctness gap about the validity of fractional moments, not a circularity of the derivation chain, and therefore is not counted here.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper adds no fitted parameters; its inputs are the FCP parameters inherited from Laskin and the choice of Lévy subordinator. The load-bearing assumptions are that the FCP pmf is valid, that the subordinator has all needed fractional moments, and that series interchanges are justified. The only invented object is the SGFBP, which is definitional rather than an independently testable entity.

assumptions (4)
  • domain assumption Laskin's FCP pmf in Eq. (1) is a genuine probability mass function for all allowed parameter values.
    The paper builds all results on this pmf from [26] without re-proving normalization or nonnegativity.
  • domain assumption The Lévy subordinator H has a density h(y,t) and finite moments E[H(t)^{θ(n+k)}] for all orders used.
    Eq. (10) and Proposition 4.1 require these moments; this is not stated for general subordinators, and it fails for α-stable subordinators beyond moment order α.
  • standard math Interchange of infinite sums with integrals and expectations is allowed.
    Used throughout Section 4 without explicit justification via dominated convergence or Tonelli's theorem.
  • standard math The generalized Mittag-Leffler asymptotic E^ζ(z) = O(|z|^{-ζ}) as |z| grows, cited from [39].
    Needed in Proposition 3.1 to justify convergence in mean of N/E[N] to 1.
invented entities (1)
  • Subordinated generalized fractional Bell polynomials (SGFBP) and numbers
    purpose: The paper defines B_SG(x,m) = sum_n n^m z(n,x) to encode TCFCP moments and to connect with Laskin's generalized fractional Bell polynomials.
    These are new mathematical definitions with no external falsifiable consequences. The claimed equality SGFBN = GFBN in Remark 5.2 appears to require E[X^{θ(n+k)}] = 1, which is not established.

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Pith. "Pith review of Fractional counting process at L\'evy times and its applications." pith.science (2026). https://pith.science/paper/LQTJH3O4

@misc{pith2026241204334,
  author       = {Pith},
  title        = {Pith review of: Fractional counting process at L\'evy times and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LQTJH3O4}},
  note         = {Machine review of arXiv:2412.04334}
}
read the original abstract

Traditionally, fractional counting processes, such as the fractional Poisson process, etc. have been defined using fractional differential and integral operators. Recently, Laskin (2024) introduced a generalized fractional counting process (FCP) by changing the probability mass function (pmf) of the time fractional Poisson process using the generalized three-parameter Mittag-Leffler function. Here, we study some additional properties for the FCP and introduce a time-changed fractional counting process (TCFCP), defined by time-changing the FCP with an independent L\'evy subordinator. We derive distributional properties such as the Laplace transform, probability generating function, the moments generating function, mean, and variance for the TCFCP. Some results related to waiting time distribution and the first passage time distribution are also discussed. We define the multiplicative and additive compound variants for the FCP and the TCFCP and examine their distributional characteristics with some typical examples. We explore some interesting connections of the TCFCP with Bell polynomials by introducing subordinated generalized fractional Bell polynomials. It is shown that the moments of the TCFCP can be represented in terms of the subordinated generalized fractional Bell polynomials. Finally, we present the application of the FCP in a shock deterioration model.

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