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Axioms 2015, 4(3), 321-344; doi:10.3390/axioms4030321

On the Fractional Poisson Process and the Discretized Stable Subordinator

1
Department of Mathematics & Computer Science, Free University Berlin, Berlin 14195, Germany
2
Department of Physics & Astronomy, University of Bologna, and INFN, Bologna 40126, Italy
*
Author to whom correspondence should be addressed.
Academic Editor: Hans J. Haubold
Received: 20 June 2015 / Revised: 27 July 2015 / Accepted: 28 July 2015 / Published: 4 August 2015
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Abstract

We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively. View Full-Text
Keywords: renewal process; Continuous Time Random Walk; erlang process; Mittag-Leffler function; wright function; fractional Poisson process; stable distributions; stable and inverse stable subordinator; diffusion limit renewal process; Continuous Time Random Walk; erlang process; Mittag-Leffler function; wright function; fractional Poisson process; stable distributions; stable and inverse stable subordinator; diffusion limit
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Gorenflo, R.; Mainardi, F. On the Fractional Poisson Process and the Discretized Stable Subordinator. Axioms 2015, 4, 321-344.

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