Some characterizations for Markov processes as mixed renewal processes

What is it about?

In this paper the class of mixed renewal processes with mixing parameter a random vector defined by Lyberopoulos and Macheras (enlarging Huang’s original class) is replaced by the strictly more comprising class of all extended mixed renewal processes 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 permanence of the Markov property under certain changes of measures.

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