Some characterizations for Markov processes as mixed renewal processes

TL;DR

This paper extends the class of mixed renewal processes by adding a second mixing parameter and investigates the conditions under which Markov processes are equivalent to mixed Poisson processes, demonstrating invariance properties and providing concrete examples.

Contribution

It introduces a broader class of extended MRPs with two mixing parameters and proves the equivalence of Markov processes and mixed Poisson processes within this class.

Findings

Markov processes are equivalent to mixed Poisson processes under certain conditions

The invariance of the Markov property under measure changes is established

Concrete examples demonstrate how to verify the Markov property using the results

Abstract

In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from \cite{lm6z3} (enlarging Huang's \cite{hu} original class) is replaced by the strictly more comprising class of all extended MRPs by adding a second mixing parameter. We prove under a mild assumption, that within this larger class the basic problem, whether every Markov process is a mixed Poisson process with a random variable as mixing parameter has a solution to the positive. This implies the equivalence of Markov processes, mixed Poisson processes, and processes with the multinomial property within this class. In concrete examples we demonstrate how to establish the Markov property by our results. Another consequence is the invariance of the Markov property under certain changes of measures.