Some characterizations for Markov processes as mixed renewal processes

Abstract

In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from [6] (enlarging Huang's [3] 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.MSC 2010: Primary 60G55 ; secondary 60K05, 28A50, 60A10, 60G05, 60J27, 91B30.

On the equivalence of various definitions of mixed poisson processes

“…Proof. The fact that tQ p θ u p θPR is a disintegration of P over P p Θ consistent with p Θ is a consequence of [10], Lemma 2.4. The equivalence piq ðñ piiiq is due to [8], Proposition 4.4.…”

“…The equivalence piq ðñ piiiq is due to [8], Proposition 4.4. Ad piq ùñ piiq : Since assertion piq holds true and tQ p θ u p θPR is a disintegration of P over P p Θ consistent with p Θ, it follows by [10], Proposition 2.2 that piiq is valid. Ad piiq ùñ piq : If piiq holds true, we get as in the proof of Theorem 2.6, piiq ùñ piq, that N has the P -Markov property; hence by [10], Theorem 2.11, we obtain piq.…”

“…Then applying [10], Proposition 2.2, we obtain piiq for tP r y u r yP r Υ :" tQ q θ u q θPR and ν :" P q Θ . Assume now in addition that Assumption 2.5 holds true.…”

“…or equivalently that N has the P -Markov property. Since N has the P -Markov property and Assumption 2.5 is valid, we may apply [10], Proposition 2.7, to obtain assertion piq.…”

“…The first one is due to Lyberopoulos and Macheras where it is proven that under the existence of an appropriate disintegration of P over P Θ a MPPpΘq can be reduced to an ordinary Poisson process under the disintegrating measures (see [8], Proposition 4.4). The second one is due to Macheras and Tzaninis where it is proven that under Assumption 2.4 within the class of mixed renewal processes, a counting process is a MPPpΘq if and only if it has the P -Markov property (see [10], Theorem 2.11). For the definition of the P -Markov property we refer to e.g.…”