Time-changed Poisson processes of order k

Abstract

In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.2010 Mathematics Subject Classification. 60G55; 60G51.

Subordinated compound Poisson processes of order k

“…On putting s = t in part (ii), we can get the expression for variance of TCPPoK(λ, H, S (k) (1)] which is mean and covariance of TCPPoK-I as discussed in (see [29,Theorem 3.2]). (1)] which is the mean and covariance of time-changed PP(λ).…”

“…A handful of researchers have also studied the distributions and processes of order k in [11,23,18]. In particular, the discrete distribution of order k, introduced by Philippou et al (see [24]), includes binomial, geometric and negative binomial distributions of order k. These distributions play an important role in several areas, such as reliability, statistics, finance, and actuarial risk analysis (see [19,29,4]).…”

“…Due to the restriction of single arrival in each inter-arrival time, the model is not suitable to use. Kostadinova and Minkova [10] introduced a Poisson process of order k (PPoK), which allows us to model the arrival in a group of size k. Recently, a time-changed version of Poisson processes of order k is studied by [29], which allows group arrivals and also accommodates the case of extreme events, which was not covered by [10]. In spite of its applicability, we observe that this model does not cover the phenomenon of underdispersion, where the variance is less than the mean (see [25,32] and references therein).…”

Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

“…To deal with dependent inter-arrival times, a generalization of Poisson processes of order k was proposed by Sengar, Maheshwari, and Upadhye (2020). These authors extended the Poisson process of order k by means of time change with a general L evy subordinator as well as an inverse L evy subordinator.…”

Skellam Type Processes of Order k and Beyond

“…Buchak and Sakhno, in [ 12 ], have also proposed the governing equations for time-fractional Skellam processes. Recently, [ 13 ] introduced time-changed Poisson process of order k , which is obtained by time changing the Poisson process of order k (see [ 14 ]) by general subordinators.…”