# Counting processes with Bernštein intertimes and random jumps

01 Dec 2015-Journal of Applied Probability (Applied Probability Trust)-Vol. 52, Iss: 4, pp 1028-1044

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TL;DR: In this article, a generalization of the space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form is introduced.

Abstract: This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.

35 citations

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TL;DR: In this paper, a generalized space-fractional Poisson process is proposed to preserve the characterizing Levy property, which is related to Prabhakar derivatives, specific convolution-type integral operators.

Abstract: The space-fractional Poisson process is a time-changed homogeneous Poisson process where the time change is an independent stable subordinator. In this paper, a further generalization is discussed that preserves the Levy property. We introduce a generalized process by suitably time-changing a superposition of weighted space-fractional Poisson processes. This generalized process can be related to a specific subordinator
for which it is possible to explicitly write the characterizing Levy measure. Connections are highlighted to Prabhakar derivatives, specific convolution-type integral operators. Finally, we study the effect of introducing Prabhakar derivatives also in time.

31 citations

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TL;DR: In this article, a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whose probabilistic ability to satisfy a suitablesystemoffractionaldifference-differential equations is considered.

Abstract: We consider a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whoseprobabilitiessatisfy a suitablesystemoffractionaldifference-differential equations. We obtain the moment generating function and the probability law of the result- ing process in terms of generalized Mittag-Leffler functions. We also discuss two equiv- alent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the countingprocess over their means tend to 1 in probability.

22 citations

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TL;DR: The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of a decreasing and constant boundary, where some closed-form results are provided and a linearly increasing boundary where an iterative procedure is proposed to compute the first-Crossing time density and survival functions.

Abstract: A compound Poisson process whose randomized time is an independent Poisson process is called a compound Poisson process with Poisson subordinator We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process The first-crossing time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing time density and survival functions

22 citations

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TL;DR: In this paper, the authors studied the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinators, which they call TCFPP-I and TC FPP-II, respectively.

Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

18 citations

##### References

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01 Jan 2010

TL;DR: In this paper, the authors present a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections, and an extensive list of complete Bernstein functions with their representations is provided.

Abstract: Bernstein functions appear in various fields of mathematics, e.g. probability theory, potential theory, operator theory, functional analysis and complex analysis- often with different definitions and under different names. Among the synonyms are `Laplace exponent' instead of Bernstein function, and complete Bernstein functions are sometimes called `Pick functions', `Nevanlinna functions' or `operator monotone functions'. This monograph- now in its second revised and extended edition- offers a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections. For the second edition the authors added a substantial amount of new material. As in the first edition Chapters 1 to 11 contain general material which should be accessible to non-specialists, while the later Chapters 12 to 15 are devoted to more specialized topics. An extensive list of complete Bernstein functions with their representations is provided.

482 citations

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TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.

Abstract: A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process. As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.

269 citations

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TL;DR: In this article, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times, which unifies the two main approaches in stochastic theory of time-fractional diffusion equations.

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

233 citations

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TL;DR: In this paper, the peculiar mechanics of the elementary particles (electrons and nuclei) that constitute ordinary matter so that the material world can have both rich variety and stability are discussed.

Abstract: Why is ordinary matter (e.g., atoms, molecules, people, planets, stars) as stable as it is? Why is it the case, if an atom is thought to be a miniature solar system, that bringing very large numbers of atoms together (say 1030) does not produce a violent explosion? Sometimes explosions do occur, as when stars collapse to form supernovae, but normally matter is well behaved. In short, what is the peculiar mechanics of the elementary particles (electrons and nuclei) that constitute ordinary matter so that the material world can have both rich variety and stability?

221 citations

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TL;DR: In this paper, a parametric family of completely random measures, which includes gamma random measures and positive stable random measures as well as inverse Gaussian measures, is defined and used in a shot-noise construction as intensity measures for Cox processes.

Abstract: A parametric family of completely random measures, which includes gamma random measures, positive stable random measures as well as inverse Gaussian measures, is defined. In order to develop models for clustered point patterns with dependencies between points, the family is used in a shot-noise construction as intensity measures for Cox processes. The resulting Cox processes are of Poisson cluster process type and include Poisson processes and ordinary Neyman-Scott processes. We show characteristics of the completely random measures, illustrated by simulations, and derive moment and mixing properties for the shot-noise random measures. Finally statistical inference for shot-noise Cox processes is considered and some results on nearest-neighbour Markov properties are given.

209 citations

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