Abstract
In this article, the compound Poisson process of order k (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinators (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that the space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the process defined in [Statist. Probab. Lett. 82 (2012), 852–858].
Highlights
The Poisson process has been the conventional model for count data analysis, and due to its popularity and applicability various researchers have generalized it in several directions; e.g., compound Poisson processes, weighted Poisson distributions, fractional versions of Poisson processes
It is worth exploring the timechange of compound Poisson process of order k (CPPoK) with a special type of Lévy subordinator known as mixture of tempered stable subordinators, its right continuous inverse, and analyze some properties of these time-changed processes
We present the mean and covariance formula for some specific cases of CPPoK(λ, H ) that we discussed in Definition 3
Summary
The Poisson process has been the conventional model for count data analysis, and due to its popularity and applicability various researchers have generalized it in several directions; e.g., compound Poisson processes, weighted Poisson distributions, fractional (time-changed) versions of Poisson processes (see [6, 21, 13, 2, 1], and references therein). We introduce the compound Poisson process of order k (CPPoK) with the help of the Poisson process of order k (PPoK) and study its distributional properties. Various versions of Poisson processes, using subordination like space and time-fractional Poisson processes have been studied in the literature (see [22, 17]) It is worth exploring the timechange of CPPoK with a special type of Lévy subordinator known as mixture of tempered stable subordinators, its right continuous inverse, and analyze some properties of these time-changed processes. These processes generalize the process discussed in [22].
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