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.
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]. Keywords Compound Poisson process of order k, mixture of tempered stable subordinators, martingale characterization 2010 MSC 60G51, 60G48
In this article, we introduce a convoluted version of Skellam process. We also give an alternate definition through its transition probabilities and discuss its distributional properties. Next, we consider the fractional version of convoluted Skellam process, which we call as fractional convoluted Skellam process (FCSP), and derive the probability generating function, mean, variance, covariance function, and establish its the long-range dependence property. Later, we introduce two time-changed variants of fractional convoluted Skellam process, in which we introduce time-change FCSP by Lévy subordinator and its first exit time, and call as TCFCSP-I and TCFCSP-II respectively. We also derive the law of iterated logarithm property for TCFCSP-I. Later we give the expression for the asymptotic behavior of moments of TCFCSP-II. MSC Classification: 60G22; 60G55
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