17 research outputs found
Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order and beyond
We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property
Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond
We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k: We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property
Fractional Generalizations of the Compound Poisson Process
This paper introduces the Generalized Fractional Compound Poisson Process
(GFCPP), which claims to be a unified fractional version of the compound
Poisson process (CPP) that encompasses existing variations as special cases. We
derive its distributional properties, generalized fractional differential
equations, and martingale properties. Some results related to the governing
differential equation about the special cases of jump distributions, including
exponential, Mittag-Leffler, Bernst\'ein, discrete uniform, truncated
geometric, and discrete logarithm. Some of processes in the literature such as
the fractional Poisson process of order , P\'olya-Aeppli process of order
, and fractional negative binomial process becomes the special case of the
GFCPP. Classification based on arrivals by time-changing the compound Poisson
process by the inverse tempered and the inverse of inverse Gaussian
subordinators are studied. Finally, we present the simulation of the sample
paths of the above-mentioned processes.Comment: 20 pages, 12 figure
Generalized Fractional Counting Process
In this paper, we obtain additional results for a fractional counting process
introduced and studied by Di Crescenzo et al. (2016). For convenience, we call
it the generalized fractional counting process (GFCP). It is shown that the
one-dimensional distributions of the GFCP are not infinitely divisible. Its
covariance structure is studied using which its long-range dependence property
is established. It is shown that the increments of GFCP exhibits the
short-range dependence property. Also, we prove that the GFCP is a scaling
limit of some continuous time random walk. A particular case of the GFCP,
namely, the generalized counting process (GCP) is discussed for which we obtain
a limiting result, a martingale result and establish a recurrence relation for
its probability mass function. We have shown that many known counting processes
such as the Poisson process of order , the P\'olya-Aeppli process of order
, the negative binomial process and their fractional versions etc. are other
special cases of the GFCP. An application of the GCP to risk theory is
discussed
Skellam Type Processes of Order K and Beyond
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.Comment: 22 pages, 1 figur
Skellam and Time-Changed Variants of the Generalized Fractional Counting Process
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 L\'evy
subordinator and its inverse. Some particular cases of these time-changed
processes are discussed by considering specific L\'evy subordinators
On the Superposition of Generalized Counting Processes
In this paper, we study the merging of independent generalized counting
processes (GCPs). First, we study the merging of finite number of independent
GCPs and then extend it to the countably infinite case. It is observed that the
merged process is a GCP with increased arrival rates. Some distributional
properties of the merged process are obtained. It is shown that a packet of
jumps arrives in the merged process according to Poisson process. An
application to industrial fishing problem is discussed