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Journal ArticleDOI

Time-changed Poisson processes of order k

TL;DR: In this article, the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse was studied, which they called TCPPoK-I and TCPPoK-II.
Abstract: In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which we call, respectively, as TCPPoK-I and TCPPoK-II, t...
Citations
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Journal ArticleDOI
22 Oct 2020-Entropy
TL;DR: In this article, the Skellam process of order k and its running average was introduced and the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes were derived.
Abstract: In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.

12 citations

Journal ArticleDOI
TL;DR: In this article, the hitting probabilities of weighted Poisson processes and their subordinated versions with different intensities were studied. And the authors analyzed the hitting probability in different weights and gave an example in the case of subordination.

4 citations

Journal ArticleDOI
TL;DR: The generalized fractional counting process (GFCP) was introduced and studied by Di Crescenzo et al. as discussed by the authors , and its covariance structure is studied, using which its long-range dependence property is established.
Abstract: In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. [8]. For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied, using which its long-range dependence property is established. It is shown that the increments of GFCP exhibit the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP), is discussed for which we obtain a limiting result and a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order k, the Pólya-Aeppli process of order k, the negative binomial process and their fractional versions etc., are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

4 citations

Journal ArticleDOI
01 May 2020
TL;DR: In this article, the compound Poisson processes of order $k$ (CPPoK) were introduced and its properties were discussed, using mixture of tempered stable subordinator and its right continuous inverse, the two subordinated CPPoK with various distributional properties were studied.
Abstract: In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.

3 citations

Journal ArticleDOI
TL;DR: In this paper, a fractional non-homogeneous Poisson Poisson process of order k and polya-aeppli Poisson Process of order K were characterized by deriving their non-local governing equations.
Abstract: We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Polya-Aeppli process of order k: We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

2 citations

References
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Journal ArticleDOI
TL;DR: Both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractionsal compound Poissonprocess are given.
Abstract: The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

19 citations

Journal ArticleDOI
TL;DR: In this paper, the authors use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory, and they find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by the two time scale subordinated of random processes.
Abstract: We use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory. Our consideration touches two physical examples in detail. First we study a temporal evolution of the species concentration for the trapping reaction in which a diffusing reactant is surrounded by a sea of randomly moving traps. The analysis is based on the random-variable formalism of anomalous diffusive processes. We find that the empirical trapping-reaction law, according to which the reactant concentration decreases in time as a product of an exponential and a stretched exponential function, can be explained by the two-time scale subordination of random processes. Another example is connected with a state equation for continuous media with memory. If the pressure and the density of a medium are subordinated in two different random processes, then the ordinary state equation becomes fractional with two time scales. This allows one to arrive at the state equation of Bagley-Torvik type.

18 citations

Posted Content
TL;DR: In this paper, the authors defined a fractional negative binomial process (FNBP) by replacing the Poisson process by a FPP in the gamma subordinated form of the negative Binomial process.
Abstract: In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $\lambda$ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to $pde$'$s$ governing the density of the FNBP and the SFPP are also investigated.

16 citations

Journal ArticleDOI
30 Jul 2018
TL;DR: In this article, a fractional negative binomial process FNBP was defined by replacing the Poisson process by a FPP in the gamma subordinated form of the NBP.
Abstract: In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Polya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

15 citations

Journal ArticleDOI
TL;DR: In this paper, a stochastic subordination of random processes in continuous media with long-term memory is considered. But it is based on the random-variable formalism of anomalous diffusive processes.

13 citations