Journal ArticleDOI

# Fractional negative binomial and pólya processes

30 Jul 2018-Vol. 38, Iss: 1, pp 77-101
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.

Topics: , , Lévy process (53%)
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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

Journal ArticleDOI
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...

13 citations

Journal ArticleDOI
Luisa Beghin1, Palaniappan Vellaisamy2Institutions (2)
Abstract: In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Levy process and the corresponding Levy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.

10 citations

Journal ArticleDOI
Abstract: The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson...

5 citations

Posted Content
13 Aug 2020-arXiv: Probability
Abstract: In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average infinite number of arrivals in any interval. We derive the marginal probabilities, governing difference-differential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes.

1 citations

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• ...th some special functions that will be required later. The Mittag-Leﬄer function Lβ(z) is deﬁned as (see [8]) (2.1) Lβ(z) = X∞ k=0 zk Γ(1+βk) , β,z∈ Cand Re(β) &gt;0. The M-Wright function Mβ(z) (see [10, 17]) is deﬁned as Mβ(z) = X∞ n=0 (−z)n n!Γ(−βn+(1− β)) = 1 π X∞ n=1 (−z)n−1 (n− 1)! Γ(βn)sin(πβn), z∈ C, 0 &lt;β&lt;1. Fractional Negative Binomial and Polya Processes 3 Let p,q∈ Z+\{0}. Also, for 1 ≤ i≤...

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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.

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### "Fractional negative binomial and pó..." refers background in this paper

• ... Wright function deﬁned in (2.5). It is also known that (see [16]) Nβ(t,λ) = N(Eβ(t),λ), where Eβ(t) is the hitting time of a stable subordinator Dβ(t). The mean and variance of FPP are given by (see [12]) ENβ(t,λ) = λtβ Γ(β+1) (3.5) ; Var(Nβ(t,λ)) = λtβ Γ(β+1) ˆ 1+ λtβ Γ(β+1) βB(β,1/2) 22β−1 − 1 ˙ (3.6) , where B(a,b) denotes the beta function. First we establish an important property of an FPP. 3...

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• ...gative binomial process, with Q(t) ∼ NB(t|η,pt), where η= α/(1+ α). Let 0 &lt;β&lt;1. A natural generalization is to consider Qβ(t) = Nβ(Γ(t),λ), where {Nβ(t)}t≥0 is a fractional Poisson process (see [12, 16]), and we call {Qβ(t)}t≥0 is a fractional negative binomial process (FNBP). We will show that this process is diﬀerent from the FNBP deﬁned in [3] and [4]. It is known that the Polya process is obtain...

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• ...t 0 &lt;β&lt;1. The fractional Poisson process (FPP) {Nβ(t,λ)}t≥0, which is a generalization of the Poisson process {N(t,λ)}t≥0 and solves the following fractional diﬀerence-diﬀerential equation (see [12, 14, 16]) Dβ t p β (n|t,λ) = −λp β (n|t,λ)+λp β (3.1) (n− 1|t,λ), for n≥ 1 Dβ t p β (0|t,λ) = −λp β (0|t,λ), where p β (n|t,λ) = P{Nβ(t,λ) = n} and (3.2) Dβ t u(t,x) = 1 Γ(1− β) Z t 0 ∂u(r,x) ∂r dr (t− r)β , ...

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