On the long-range dependence of fractional Poisson and negative binomial processes
TL;DR: In this article, the short-range dependence (SRD) property of the increments of the fractional Poisson process was discussed, and it was shown that fractional negative binomial process (FNBP) has the same property.
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).
Citations
More filters
Posted Content•
TL;DR: In this article, a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator is introduced and an integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained.
Abstract: We introduce and study a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order $k$ (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of the FSPoK which are utilized to establish its long-range dependence property. Later, we considered two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent L\'evy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.
1 citations
TL;DR: In this paper, the optimal retentions for an insurer with a compound fractional Poisson surplus and a layer reinsurance treaty were studied under the criterion of maximizing the adjustment coefficient.
Abstract: In this paper, we study the optimal retentions for an insurer with a compound fractional Poisson surplus and a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained. It is demonstrated that the optimal retention vector and the maximal adjustment coefficient are not only closely related to the parameter of the fractional Poisson process, but also dependent on the time and the claim intensity, which is different from the case in the classical compound Poisson process. Numerical examples are presented to show the impacts of the three parameters on the optimal results.
1 citations
TL;DR: In this article , the authors defined a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations, and obtained its one-dimensional distribution by using the Laplace transform of its state probabilities.
Abstract: In this article, we define a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations. Next, we obtain its one-dimensional distribution by using the Laplace transform of its state probabilities. Various distributional properties, such as probability generating function, moment generating function and moments, are derived. A special case of CFPPoK, (say) convoluted Poisson process of order k (CPPoK) is studied and also established Martingale characterization for CPPoK. We further derive the covariance structure of CFPPoK and investigate the long-range dependence property.
TL;DR: The marginal probabilities, Lévy measures, governing difference-differential equations of the introduced processes, and the results generalize the Skellam process and running average of Poisson process in several directions.
Abstract: In this article, we introduce Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular we discuss space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Poisson 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 Skellam process and running average of Poisson process in several directions.
TL;DR: In this article , the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator, and it is observed that the GFSP is a Skellham type version of the generalized fractional counting process (GFCP), which is a fractional variant of the GCP.
Abstract: In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Lévy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Lévy subordinators.
References
More filters
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
TL;DR: The authors provides a survey of the work that has been done in financial econometrics in the past decade, establishing a set of stylized facts that are characteristics of financial series and then detailing the range of techniques that have been developed to model series which possess these characteristics.
739 citations
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