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

Abstract: In this paper, we obtain some additional probabilistic properties of the renewal process ${\hat{N}α(t)}{t\ge0}$, $0<α\le 1$ introduced by Beghin and Orsingher (2010). A time-changed relationship connecting ${\hat{N}α(t)}{t\ge0}$ with its special case ${\hat{N}(t)}{t\ge0}$ by means of the random time process ${T{2α}(t)}{t>0}$ 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 ${\hat{N}α(t)}{ t \ge 0}$ 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 ${\hat{N}α(t)}{ t \ge 0}$ and discuss an application to ruin theory. Later, we consider two time-changed versions of ${\hat{N}α(t)}_{ t \ge 0}$ which are obtained by time-changing it with an independent Lévy subordinator and its inverse. Some distributional properties and examples are discussed for these time-changed processes.

• The paper demonstrates that renewal processes with Mittag-Leffler waiting times exhibit non-infinite divisibility and derive explicit pmf and pgf expressions using three-parameter Mittag-Leffler functions.
• The paper establishes subordination and time-change relationships with Lévy subordinators, extending classical renewal theory to encapsulate heavy-tailed and anomalous diffusion behaviors.
• The paper shows that scaling limits lead to convergence of normalized counts, providing valuable asymptotic insights for risk assessment and stochastic modeling.

Renewal Processes with Mittag-Leffler Waiting Times: Probabilistic Properties and Time-Changed Generalizations

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Background and Motivation

This paper investigates the renewal process $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$, parameterized by $0<\alpha\le1$, introduced by Beghin and Orsingher (2010). While the classical Poisson process has exponentially distributed interarrival times, the process in focus is governed by Mittag-Leffler waiting times and higher-order fractional differential equations. This structure facilitates modeling phenomena with heavy-tailed interarrival distributions and anomalous diffusion, which are prominent in aging, heterogeneous materials, finance, and reliability theory.

The process satisfies a system of fractional differential equations and exhibits intricate probabilistic structure, including relationships with the fractional Poisson process (TFPP), time-changing mechanisms via subordinators and their inverses, as well as connections to ruin theory in insurance mathematics.

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Main Results and Analytical Framework

Distributional and Moment Structures

The paper thoroughly details distributional properties of $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$, including explicit expressions for the probability mass function (pmf) and probability generating function (pgf). Both expressions exploit the three-parameter Mittag-Leffler functions, capturing the higher-order renewal structure. The process is shown to have variance and higher moments derived via Laplace transforms and Mellin-Barnes representations, with the factorial moments $\hat{\psi}_\alpha(r,t)$ given as

$$\hat{\psi}_\alpha(r,t)=r!\lambda^{2r}t^{2\alpha r}E^r_{\alpha,2\alpha r+1}(-2\lambda t^\alpha).$$

Covariances (in the Laplace domain) follow from renewal theory formulations and are intricately dependent on the Laplace transforms of the interarrival time densities.

Subordination and Time-Change Relationships

A central result is the establishment of subordination and time-change links:

• $$\hat{N}_{\alpha}(t)\overset{d}{=}\hat{N}(T_{2\alpha}(t))$$, where $T_{2\alpha}(t)$ solves fractional diffusion equations, generalizing reflecting Brownian motion for specific $\alpha$.
• For subordinators $D_f(t)$, time-changed versions $\hat{N}_\alpha^{f}(t):=\hat{N}_\alpha(D_f(t))$ are analyzed; explicit pmfs and pgfs are provided in terms of moments of $0<\alpha\le1$0, accommodating gamma, tempered stable, and inverse Gaussian cases.

The equivalence in distribution between $0<\alpha\le1$1 and renewal processes with generalized Mittag-Leffler interarrival times (per Cahoy and Polito, 2012) is demonstrated analytically.

Scaling Limits and Asymptotic Behaviors

The process exhibits a convergence property where scaled ratios $0<\alpha\le1$2 and their powers approach unity in probability as $0<\alpha\le1$3, indicating concentration phenomena akin to cutoff behaviors in Markov processes. Scaling limits (as $0<\alpha\le1$4) yield limiting distributions involving stable subordinators, with moments dependent on the order parameter $0<\alpha\le1$5. Explicitly, for large $0<\alpha\le1$6,

$0<\alpha\le1$7

where $0<\alpha\le1$8 is the normalized count, and its limiting distribution is described via $0<\alpha\le1$9 (the stable subordinator density).

Infinite Divisibility

A bold claim in the paper is that the one-dimensional distributions of $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$0 are not infinitely divisible—a departure from classical renewal processes. This is established by contradiction, noting that the involved limiting random variable $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$1 (inverse stable subordinator) fails to be infinitely divisible, as recently proved [Kumar & Nane, 2018].

Compound and Risk Processes

A compound version $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$2 is examined for i.i.d. claims $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$3. Resulting mean, variance, and mgf are obtained via Wald’s identities. Applications to risk and insurance theory include:

• Explicit covariance expressions for risk reserve processes.
• Density and asymptotic formulas for ruin time and ruin probabilities under both exponential and subexponential claim size regimes.

For subexponential claims, the asymptotic finite-time ruin probability is shown to satisfy

$\{\hat{N}_{\alpha}(t)\}_{t\ge0}$4

where $\{\hat{N}_{\alpha}(t)\}_{t\ge0}$5 is the tail of the claim size distribution.

Time-Change by Lévy Subordinators and Inverses

The processes time-changed by Lévy subordinators and their inverses are analyzed, yielding new renewal structures. Specifically:

• The pmf, pgf, mean, and variance for time-changed processes are expressed in terms of moments of the subordinator and/or its inverse.
• Laws of the iterated logarithm and scaling limits are established for these processes under regular variation assumptions.

Bivariate distributions for time-changed cases are derived, generalizing those for the basic renewal process, utilizing convolution and conditioning arguments typical in renewal theory.

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Practical and Theoretical Implications

The findings broaden the class of renewal and counting processes available for modeling heavy-tailed, anomalously diffusive and aging phenomena, with practical utility in:

• Risk and insurance mathematics: non-Poissonian claim processes, explicit formulas for ruin probabilities reflect more realistic heavy-tailed claim arrivals.
• Finance and stochastic modeling: time-changed processes accommodate stochastic volatility and random time deformation effects.
• Statistical physics and hydrology: connections to fractional diffusion and random walks underscore applications to transport in heterogeneous media.

The explicit analytical forms for moments, covariances, and distributional quantities enable feasible parameter estimation and simulation-based inference in applied domains.

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Future Research Directions

Potential avenues include:

• Parameter estimation and statistical inference for Mittag-Leffler renewal processes and their time-changed variants.
• Further exploration of processes with generalized or variable-order fractional derivatives, as initiated by Beghin et al. (2026).
• Extensions to multivariate renewal and risk processes, coupling multiple heavy-tailed claim streams and examining systemic risk.
• Investigation of ergodicity, recurrence, and transient behaviors in higher-order or non-local fractional renewal systems.

These prospects coincide with ongoing research in fractional calculus, stochastic processes, insurance mathematics, and applied probability.

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Conclusion

The paper systematically characterizes the probabilistic, asymptotic, and operational properties of renewal processes with Mittag-Leffler waiting times, their compound and risk applications, and their time-changed generalizations via subordinators and inverse subordinators. Explicit analytical forms and rigorous limit behaviors underscore the richness and applicability of fractional renewal processes in modeling complex stochastic systems, with bold claims regarding non-infinite divisibility and operationally relevant ruin probabilities (2604.07275).