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A fractional generalization of the Poisson processes. (English) Zbl 1087.60064
The concept of renewal process has been developed as a stochastic model for describing the class of counting processes for which the times between successive events (waiting times) are independent and identically distributed, non-negative random variables, obeying a given probability law. For a renewal process having waiting times $$T_1$$, $$T_2$$,…, let $$t_0=0$$, $$t_k=\sum_{j=1}^k T_j$$, $$k\geq1$$ ($$t_k$$ is the time of the $$k$$th renewal). The process is specified if we know the probability law for the waiting times ($$\phi(t)$$ is its density function and $$\Phi(t)$$ the distribution function of the waiting times). $$\Psi(t)= 1 - \Phi(t)$$ is the survival probability. A relevant quantity is the counting function $$N(t)=\max\{k: t_k\leq t$$, $$k=0,1,2,\dots\}$$. In particular, $$\Psi(t)=P(N(t)=0)$$. The most celebrated renewal process is the Poisson process characterized by a waiting time density function of exponential type. The survival probability for the Poisson renewal process obeys the ordinary differential equation of relaxation type: $${{d\Psi(t)}/{dt}} = -\,\Psi(t)$$, $$t\geq 0$$ and $$\Psi(0^+)=0$$. A “fractional” generalization of the Poisson renewal process is simply obtained by generalizing the differential equation $${{d\Psi(t)}/{dt}} = - \Psi(t)$$ replacing there the first derivative with the integro-differential operator $${}_tD_*^{\beta}$$ that is interpreted as the fractional derivative of order $$\beta$$, $$0<\beta\leq 1$$. In the present paper the authors analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. The Mittag-Leffler function of parameter $$\beta$$ is defined in the complex plane $$z\in\mathcal C$$ by the power series: $$E_{\beta}(z) = \sum_{n=0}^{\infty}{{z^n}/{\Gamma(\beta\,n+1)}}$$. It turns out to be an entire function of order $$\beta$$ which reduces for $$\beta=1$$ to $$\exp(z)$$. The solution of the equation $${}_tD_*^{\beta}\Psi(t) = -\Psi(t)$$ is known to be $$\Psi(t) = E_{\beta}(-t^{\beta})$$, $$t\geq0$$ and $$0<\beta\leq 1$$, see for example R. Gorenflo and F. Mainardi [in: Fractals and fractional calculus in continuum mechanics, 223–276, (1997)]. In contrast to Poisson case $$\beta=1$$, in the case $$0<\beta<1$$ for large $$t$$ the functions $$\Psi(t)$$ and $$\phi(t)$$ no longer decay exponentially but algebraically. As a consequence of the power-law asymptotics the process turns to be no longer Markovian but of long-memory type. However, for $$0<\beta<1$$ both functions $$\Psi(t)$$ and $$\phi(t)$$ keep the “completely monotonic” character of the Poisson case. Completely monotonicity of the function $$F$$ means $$(-1)^n\,{{d^n F(t)}/{dt^n}} \geq0$$ for $$n=0,1,2,\dots$$ and $$t\geq0$$ or equivalently, representability of $$F(t)$$ as real Laplace transform of nonnegative generalized functions (or measures), see Gorenflo and Mainardi [loc. cit.].
The paper is organized as follows: Section 1 recalls some notions from renewal processes. Section 2 considers the classical case, the Poisson renewal process. Section 3 deals with the Mittag-Leffler generalization of the Poisson renewal process and asymptotics for functions $$\Psi(t)$$ and $$\phi(t)$$ for $$t\to \infty$$. In Section 4 the authors have obtained that the Mittag-Leffler probability distribution is the limiting distribution for the thinning procedure of a generic renewal process with waiting time density of power law character. Section 5 deals with renewal processes by reward, that means that every renewal instant as space-like variable makes a random jump from its previous position to a new point in space. The stochastic evolution of the space variable in time is modelled by an integro-differential equation which, by containing a time fractional derivative, can be considered as the fractional generalization of the classical Kolmogorov-Feller equation of the compound Poisson process.

##### MSC:
 60K05 Renewal theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 30D10 Representations of entire functions of one complex variable by series and integrals
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