scholarly journals On the long-range dependence of fractional Poisson and negative binomial processes

2016 ◽  
Vol 53 (4) ◽  
pp. 989-1000 ◽  
Author(s):  
A. Maheshwari ◽  
P. Vellaisamy
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Short Range ◽  
Negative Binomial ◽  
Stationary Process ◽  
Long Range Dependence ◽  
Fractional Poisson Process ◽  
Poissonian Noise ◽  
Negative Binomial Process ◽  
Binomial Process

Abstract We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

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 Pólya 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.

scholarly journals Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

2014 ◽  
Vol 51 (3) ◽  
pp. 727-740 ◽  
Author(s):  
Romain Biard ◽  
Bruno Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Risk Model ◽  
Nonstationary Process ◽  
Long Range Dependence ◽  
Insurance Company ◽  
Surplus Process ◽  
Fractional Poisson Process ◽  
Renewal Risk ◽  
Dependence Property

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

scholarly journals Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

2014 ◽  
Vol 51 (03) ◽  
pp. 727-740 ◽  
Author(s):  
Romain Biard ◽  
Bruno Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Risk Model ◽  
Nonstationary Process ◽  
Long Range Dependence ◽  
Insurance Company ◽  
Surplus Process ◽  
Fractional Poisson Process ◽  
Renewal Risk ◽  
Dependence Property

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

A generalised Dickman distribution and the number of species in a negative binomial process model

2021 ◽  
Vol 53 (2) ◽  
pp. 370-399
Author(s):  
Yuguang Ipsen ◽  
Ross A. Maller ◽  
Soudabeh Shemehsavar
Keyword(s):  
Poisson Process ◽  
Process Model ◽  
Negative Binomial ◽  
Sample Distribution ◽  
Large Sample ◽  
Number Of Species ◽  
Dirichlet Model ◽  
Negative Binomial Process ◽  
Binomial Process ◽  
Two Parameter

AbstractWe derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (04) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pk f of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tk f of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Correlation models with long-range dependence

2002 ◽  
Vol 39 (2) ◽  
pp. 370-382 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Time Series ◽  
Long Range ◽  
Long Memory ◽  
Short Range ◽  
Sufficient Conditions ◽  
Long Range Dependence ◽  
Simple Method ◽  
Summable Sequence ◽  
Correlation Models ◽  
Necessary And Sufficient

This paper is concerned with the correlation structure of a stationary discrete time-series with long memory or long-range dependence. Given a sequence of bounded variation, we obtain necessary and sufficient conditions for a function generated from the sequence to be a proper correlation function. These conditions are applied to derive various slowly decaying correlation models. To obtain correlation models with short-range dependence from an absolutely summable sequence, a simple method is introduced.

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