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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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