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...
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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
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
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
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
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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TL;DR: In this paper, a continuous time random walk (CTRW) with an event time distribution governed by ψ(t) ∼ Aβt−1−β is used to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments.
Abstract: [1] We present a physical model to explain the behavior of long-term, time series measurements of chloride, a natural passive tracer, in rainfall and runoff in catchments [Kirchner et al., Nature, 403(524), 2000]. A spectral analysis of the data shows the chloride concentrations in rainfall to have a white noise spectrum, while in streamflow, the spectrum exhibits a fractal 1/f scaling. The empirically derived distribution of tracer travel times h(t) follows a power-law, indicating low-level contaminant delivery to streams for a very long time. Our transport model is based on a continuous time random walk (CTRW) with an event time distribution governed by ψ(t) ∼ Aβt−1−β. The CTRW using this power-law ψ(t) (with 0 < β < 1) is interchangeable with the time-fractional advection-dispersion equation (FADE) and has accounted for the universal phenomenon of anomalous transport in a broad range of disordered and complex systems. In the current application, the events can be realized as transit times on portions of the catchment network. The travel time distribution is the first passage time distribution F(t;l) at a distance l from a pulse input (at t = 0) at the origin. We show that the empirical h(t) is the catchment areal composite of F(t;l) and that the fractal 1/f spectral response found in many catchments is an example of the larger class of transport phenomena cited above. The physical basis of ψ(t), which determines F(t;l), is the origin of the extremely long chemical retention times in catchments.
178 citations
TL;DR: In this article, the scaling limits of continuous time random walks were shown to be equivalent to the hitting time process of a classical stable subordinator, and a close and unexpected connection between these two classes of processes and an equivalence between two families of partial differential equations.
Abstract: A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involves subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes and, consequently, an equivalence between these two families of partial differential equations.
121 citations
TL;DR: In this paper, the authors introduce the space-fractional Poisson process whose state probabilities p, t, t > 0, � 2 (0,1), are governed by the equations (d/dt)pk(t) = � � (1 B)p � (t), where (B) is the fractional difference operator found in the study of time series analysis.
110 citations
TL;DR: In this paper, the authors introduce a class of interesting stochastic processes based on Brownian-time processes, which are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion.
Abstract: We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE’s). We then study their exit problem as they exit nice domains in $\mathbb{R}^d$ , and connect it to elliptic PDE’s. We show that these processes have the peculiar property that they solve fourth order parabolic PDE’s, but their exit distribution—at least in the standard Brownian time process case—solves the usual second order Dirichlet problem. We recover fourth order PDE’s in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.
104 citations
TL;DR: In this paper, a formal estimation procedure for parameters of the fractional Poisson process (fPp) is proposed to make the fPp model more flexible by permitting non-exponential, heavy-tailed distributions of interarrival times and different scaling properties.
102 citations