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On the Long-range Dependence of Fractional Poisson and Negative Binomial Processes
TLDR
It is established that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property.Abstract:
We study the long-range dependence (LRD) of the increments of the fractional Poisson process (FPP), the fractional negative binomial process (FNBP) and the increments of the FNBP. We first point out an error in the proof of Theorem 1 of Biard and Saussereau (2014) and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index $\beta$ satisfies $0<\beta<\frac{1}{3}$. We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.read more
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Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse
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.
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A fractional counting process and its connection with the Poisson process
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.
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Limit theorems for the fractional non-homogeneous Poisson process
TL;DR: Both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractionsal compound Poissonprocess are given.
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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.
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Non-homogeneous space-time fractional Poisson processes
TL;DR: The space-time fractional Poisson process (STFPP) as mentioned in this paper is a generalization of the TFPP and the space fractional poisson process, defined by Orsingher and Poilto (2012).
References
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The Fractional Poisson Process and the Inverse Stable Subordinator
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.
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Fractional Poisson processes and related planar random motions
Luisa Beghin,Enzo Orsingher +1 more
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
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Long-memory processes in ozone and temperature variations at the region 60° S–60° N
TL;DR: In this article, global column ozone and tropospheric temperature observations made by ground-based (1964-2004) and satellite-borne (1978−2004) instrumentation are analyzed and found to be positively correlated to those in larger time-intervals in a power-law fashion.
Journal Article
A fractional generalization of the Poisson processes
TL;DR: In this article, a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function is analyzed, and it is shown that this distribution plays a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time.
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A fractional generalization of the Poisson processes
TL;DR: In this article, a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function is analyzed, and it is shown that this distribution plays a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time.