Subordinated Compound Poisson processes of order $k$

TL;DR

This paper introduces and analyzes compound Poisson processes of order k, exploring their properties, subordinated versions, and fractional generalizations, thereby extending existing models in stochastic process theory.

Contribution

The paper presents new subordinated CPPoK models using mixture of tempered stable subordinators and develops fractional versions, broadening the scope of compound Poisson processes.

Findings

Properties of CPPoK are characterized.

Subordinated CPPoK models are constructed using MTSS.

Fractional versions of CPPoK are derived.

Abstract

In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.