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On the Long-range Dependence of Fractional Poisson and Negative Binomial Processes
TL;DR: 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.
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TL;DR: In this article, it was shown that the mixed fractional Poisson process exhibits the long-range dependence (LRD) property and established an asymptotic result for the covariance of inverse mixed stable subordinator.
Abstract: In this paper, we show that the mixed fractional Poisson process (MFPP) exhibits the long-range dependence (LRD) property. It is proved by establishing an asymptotic result for the covariance of inverse mixed stable subordinator. Also, it is shown that the increments of the MFPP, namely, the mixed fractional Poissonian noise (MFPN) has the short-range dependence (SRD) property.
3 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, the authors define the delayed Levy-driven continuous-time autoregressive process via the inverse of the stable subordinator and derive correlation structure for the observed non-stationary delayed Levy drive.
Abstract: We define the delayed Levy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Levy-dri...
3 citations
Cites methods from "On the Long-range Dependence of Fra..."
...Long-range dependence property in the form of Definition 5.1 was first used in Maheshwari & Vellaisamy (2016) and Maheshwari & Vellaisamy (2017)....
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TL;DR: In this article, the authors studied the fractional Poisson process (FPP) time-changed by an independent L'evy subordinator and the inverse of the L\'evy sub-subordinator.
Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent L\'evy subordinator and the inverse of the L\'evy 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. Its bivariate distributions and also the governing difference-differential equation are derived. Some specific examples for both the processes are discussed. Finally, we present the simulations of the sample paths of these processes.
2 citations
Cites background or methods from "On the Long-range Dependence of Fra..."
...1 of [23] to the subordinator {Df (t)}t≥0....
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...We now present the definition (see [13, 23]) that will be used in this paper....
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TL;DR: In this paper, the authors introduced and studied a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities.
Abstract: In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Levy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.
2 citations
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01 Jan 1950
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Abstract: Office hours: MWF, immediately after class or early afternoon (time TBA). We will cover the mathematical foundations of probability theory. The basic terminology and concepts of probability theory include: random experiments, sample or outcome spaces (discrete and continuous case), events and their algebra, probability measures, conditional probability A First Course in Probability (8th ed.) by S. Ross. This is a lively text that covers the basic ideas of probability theory including those needed in statistics. Theoretical concepts are introduced via interesting concrete examples. In 394 I will begin my lectures with the basics of probability theory in Chapter 2. However, your first assignment is to review Chapter 1, which treats elementary counting methods. They are used in applications in Chapter 2. I expect to cover Chapters 2-5 plus portions of 6 and 7. You are encouraged to read ahead. In lectures I will not be able to cover every topic and example in Ross, and conversely, I may cover some topics/examples in lectures that are not treated in Ross. You will be responsible for all material in my lectures, assigned reading, and homework, including supplementary handouts if any.
10,221 citations
Additional excerpts
...The negative binomial process {Q(t, λ)}t≥0 = {N(Y (t), λ)}t≥0 is a subordinated Poisson process (see [6, 9]) with P[Q(t, λ) = n] = δ(n|α, pt, λ) = ( n+ pt− 1 n )...
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TL;DR: In this paper, a Monte-Carlo analysis of stock market returns was conducted and it was found that not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute return also has quite high autocorrelation for long lags.
3,462 citations
"On the Long-range Dependence of Fra..." refers background in this paper
...It has applications to several areas, such as Internet data traffic modeling [8], finance [3], econometrics [14], hydrology [4, p....
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TL;DR: The authors outline LRD findings in network traffic and explore the current lack of accuracy and robustness in LRD estimation and present recent evidence that packet arrivals appear to be in agreement with the Poisson assumption in the Internet core.
Abstract: Self-similarity and scaling phenomena have dominated Internet traffic analysis for the past decade. With the identification of long-range dependence (LRD) in network traffic, the research community has undergone a mental shift from Poisson and memory-less processes to LRD and bursty processes. Despite its widespread use, though, LRD analysis is hindered by the difficulty of actually identifying dependence and estimating its parameters unambiguously. The authors outline LRD findings in network traffic and explore the current lack of accuracy and robustness in LRD estimation. In addition, they present recent evidence that packet arrivals appear to be in agreement with the Poisson assumption in the Internet core.
316 citations
"On the Long-range Dependence of Fra..." refers background in this paper
...It has applications to several areas, such as Internet data traffic modeling [8], finance [3], econometrics [14], hydrology [4, p....
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TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.
302 citations
"On the Long-range Dependence of Fra..." refers background or methods in this paper
...The fractional Poisson process (FPP) {Nβ(t, λ)}t≥0, which is a generalization of the Poisson process {N(t, λ)}t≥0, is defined as the stochastic process whose pβ (n | t, λ) = P[Nβ(t, λ) = n] satisfies (see [10], [12], and [13])...
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...Let {Nβ(t)}t≥0 be a fractional Poisson process (see [10]), where we call β the fractional index....
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...The mean and the variance of the FPP are given by (see [10])...
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