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arxiv: 2604.07275 · v1 · submitted 2026-04-08 · 🧮 math.PR

Some probabilistic properties and time-changed versions of a renewal process based on Mittag-Leffler waiting times

Mostafizar Khandakar , Bratati Pal This is my paper

Pith reviewed 2026-05-10 17:48 UTC · model grok-4.3

classification 🧮 math.PR
keywords renewal processMittag-Leffler distributiontime-changed processfractional diffusionLevy subordinatorruin theoryinfinite divisibilitycompound process
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The pith

A Mittag-Leffler renewal process admits explicit moments, a fractional time-change link, and ruin-theory applications.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives additional distributional properties for the renewal counting process driven by independent Mittag-Leffler waiting times. It computes the variance, factorial moments, moment generating function, and Laplace-domain covariance, then shows that the process and its powers normalized by their means converge in probability to one. A random time change connects the fractional-alpha version to the ordinary renewal process through a clock whose law solves a fractional diffusion equation. The authors also obtain an integral expression for the bivariate distribution, prove the marginal laws are not infinitely divisible, study the compound version with an application to ruin theory, and introduce two further time-changed versions driven by Lévy subordinators and their inverses.

Core claim

The renewal process with Mittag-Leffler interarrivals has computable moments and covariance; its normalized versions converge in probability to one; its one-dimensional distributions are not infinitely divisible; it satisfies the time-change relation connecting it to the standard case via the random clock T_{2α}(t) tied to a fractional diffusion equation; and two additional versions arise by subordinating with an independent Lévy process and its inverse.

What carries the argument

The Mittag-Leffler renewal counting process defined by i.i.d. Mittag-Leffler waiting times, together with the random time process T_{2α}(t) whose distribution is governed by a fractional diffusion equation that realizes the time change.

If this is right

  • The explicit moment and covariance formulas can be plugged directly into calculations for the compound process.
  • The time-change relation allows the fractional process to be simulated from an ordinary renewal process driven by the auxiliary clock.
  • The non-infinite divisibility limits the ways the process can be embedded inside a Lévy or Markov framework.
  • The ruin-theory application yields explicit expressions for ruin probabilities under Mittag-Leffler interclaim times.

Where Pith is reading between the lines

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

  • The construction supplies concrete, simulable examples of point processes with controllable memory that sit between ordinary renewal and fractional Poisson models.
  • Empirical tests could compare the predicted convergence rates or covariance structure against data from systems whose inter-event times exhibit Mittag-Leffler tails.
  • Further subordinators beyond Lévy could be substituted to generate still richer families of non-Markovian counting processes.

Load-bearing premise

The waiting times stay independent and identically distributed as Mittag-Leffler for every alpha in (0,1] and obey the classical renewal axioms without extra regularity requirements that might fail in the fractional regime.

What would settle it

A direct numerical or analytic check for a specific alpha in (0,1) showing that the ratio of the process (or its power) to its mean fails to converge in probability, or that the derived Laplace-domain covariance expression does not hold.

Figures

Figures reproduced from arXiv: 2604.07275 by Bratati Pal, Mostafizar Khandakar.

Figure 1
Figure 1. Figure 1: The instants of occurrence of the events and the related waiting times. τˆ α,m 1 > t − Tˆα l − τˆ α,l 1 − τˆ α,l+1 m−l−1 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Waiting times between events for n = l. 0 s t ←−−−−−−−→ τ (1) l ←−−−−−−−−−−−−−−−−−−→ τ (n−l−1) l+1 ←−−−−−−−→ τ (1) n Tl Tl+1 Tn Tn+1 [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Waiting times between events for n ≥ l + 1. Proof. For n = l, we have (see [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
read the original abstract

In this paper, we obtain some additional probabilistic properties of the renewal process $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$, $0<\alpha\le 1$ introduced by Beghin and Orsingher (2010). A time-changed relationship connecting $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$ with its special case $\{\hat{N}(t)\}_{t\ge0}$ by means of the random time process $\{T_{2\alpha}(t)\}_{t>0}$ whose distribution is related to a fractional diffusion equation is established. We compute its various distributional properties such as the variance, factorial moments, moment generating function, moments, covariance in the Laplace domain, etc. We show that the ratios given by $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ and its power over their means tend to $1$ in probability. Moreover, we derive an integral form of its bivariate distribution and describe the scaling limits of its marginal distributions. It is also shown that its one-dimensional distributions are not infinitely divisible. Furthermore, we study the compound version of $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ and discuss an application to ruin theory. Later, we consider two time-changed versions of $\{\hat{N}_{\alpha}(t)\}_{ t \ge 0}$ which are obtained by time-changing it with an independent L\'evy subordinator and its inverse. Some distributional properties and examples are discussed for these time-changed processes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript extends the renewal counting process {N̂_α(t)} t≥0 (0<α≤1) with i.i.d. Mittag-Leffler interarrival times introduced by Beghin and Orsingher (2010). It derives Laplace-domain expressions for moments, variance, factorial moments, MGF, and covariance; establishes a time-change representation linking N̂_α(t) to the α=1 case via the inverse 2α-stable subordinator T_{2α}(t) whose density solves a fractional diffusion equation; proves convergence in probability of N̂_α(t)/E[N̂_α(t)] and [N̂_α(t)]^p to 1; obtains an integral form of the bivariate distribution and scaling limits; shows the marginal laws are not infinitely divisible; studies the compound process and its ruin-theory application; and examines two further time-changed versions driven by an independent Lévy subordinator and its inverse.

Significance. If the derivations are correct, the work supplies explicit distributional tools and a time-change identity that connect fractional renewal theory to stable subordinators and fractional PDEs. These may facilitate simulation and analysis in applications such as ruin probabilities. The use of Laplace-transform methods that remain valid when the mean interarrival time is infinite is a technical strength, as is the explicit treatment of the compound and doubly time-changed extensions.

major comments (2)
  1. [§3] §3 (time-change relation): the statement that N̂_α(t) equals N̂(T_{2α}(t)) in distribution requires an explicit verification that the inverse subordinator T_{2α} is independent of the underlying renewal process and that the composition preserves the Mittag-Leffler interarrival law; the current outline does not display the necessary conditioning argument or Laplace-transform identity that would confirm this for 0<α<1.
  2. [§4.2] §4.2 (non-infinite divisibility): the proof that the one-dimensional distributions are not infinitely divisible appears to rest on the form of the Laplace transform of the waiting-time distribution; it should be checked whether the same argument continues to hold after the time-change by T_{2α}, or whether the claim is restricted to the original process.
minor comments (3)
  1. Notation for the inverse subordinator T_{2α}(t) is introduced without a displayed definition of its Laplace transform or density; adding the explicit expression (even if standard) would improve readability.
  2. [§4.1] The convergence-in-probability statements in §4.1 are stated for the ratios over their means; the paper should clarify whether the means are finite for all α∈(0,1] or only for α=1.
  3. Several Laplace-domain expressions are given without the corresponding inversion formulas or numerical illustrations; a short table of explicit moments for selected α would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments below and will revise the paper accordingly to provide the requested clarifications.

read point-by-point responses

  1. Referee: [§3] §3 (time-change relation): the statement that N̂_α(t) equals N̂(T_{2α}(t)) in distribution requires an explicit verification that the inverse subordinator T_{2α} is independent of the underlying renewal process and that the composition preserves the Mittag-Leffler interarrival law; the current outline does not display the necessary conditioning argument or Laplace-transform identity that would confirm this for 0<α<1.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will add a short subsection detailing the argument: the inverse 2α-stable subordinator is constructed independently of the base renewal process {N̂(t)}, and we condition on T_{2α}(t)=s. The Laplace transform of the counting process then reduces via the known transform of the inverse subordinator to the Mittag-Leffler interarrival law, confirming the equality in distribution for 0<α<1. revision: yes

  2. Referee: [§4.2] §4.2 (non-infinite divisibility): the proof that the one-dimensional distributions are not infinitely divisible appears to rest on the form of the Laplace transform of the waiting-time distribution; it should be checked whether the same argument continues to hold after the time-change by T_{2α}, or whether the claim is restricted to the original process.

    Authors: The non-infinite-divisibility result applies only to the original process {N̂_α(t)}. Its proof uses the specific Laplace transform of the Mittag-Leffler interarrival times; after the time change the marginal laws differ, and we make no claim about infinite divisibility for the time-changed version. We will insert an explicit clarifying sentence to this effect. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines the renewal process via i.i.d. Mittag-Leffler interarrival times (extending the 2010 Beghin-Orsingher construction) and derives Laplace-domain moments, time-change relations via the inverse 2α-stable subordinator, factorial moments, covariance, scaling limits, non-infinite-divisibility, and compound-process properties using standard generating-function and Laplace-transform techniques. These steps rely on the independence/stationarity axioms of renewal theory and the known Mittag-Leffler Laplace transform; none reduce by construction to a fitted parameter inside the paper, a self-citation chain, or an ansatz smuggled via prior work by the same authors. The 2010 citation supplies only the base definition, not a load-bearing uniqueness theorem or fitted input. All claimed results remain independently verifiable from the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior definition of the Mittag-Leffler renewal process and classical renewal axioms; no new free parameters beyond the given range for α or invented entities are introduced in the abstract.

free parameters (1)
  • alpha
    Fractional parameter 0 < α ≤ 1 that shapes the Mittag-Leffler waiting-time distribution; treated as given rather than fitted inside this work.
axioms (1)
  • domain assumption Waiting times are i.i.d. with Mittag-Leffler distribution and the process satisfies the standard renewal counting definition.
    Invoked when extending the 2010 process to new properties and time changes.

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