Some characterizations of mixed Poisson processes

Some characterizations for Markov processes as mixed renewal processes

“…e.g [8], Theorem 2.3.4). Thus, applying [4], Proposition 4.4, we deduce that N is a MPPp q Θq. l Remarks 2.8 (a) Assumption 2.6 is a modification of Huang's Assumption p˚q since there it is assumed the stronger condition 0 ă F 1 θ ptq ă C for any t ą 0 and θ R L, where C is a positive constant, in the place of 0 ă F 1 θ ptq ă Cphpθqq for any t ą 0 and θ R L for C P L 1 pP hpΘq q in Assumption 2.6.…”

“…Moreover, applying Lemma 3.5, of [4] and (a) after some manipulation as in the proof of (c) we obtain…”

On the equivalence of various definitions of mixed poisson processes

“…Proof. First note that implication piiiq ùñ piq is immediate by Proposition 4.4 of [8], while the implication piq ùñ pivq follows by an easy computation. Assume now that P is perfect and Σ is countably generated.…”

“…Assume now that P is perfect and Σ is countably generated. Then the equivalence piq ðñ piiiq follows by [8], Proposition 4.4, since under this assumption for any realvalued random variable q Θ on Ω there always exists a disintegration tQ q θ u q θPR of P over P q Θ consistent with q Θ (see [3], Theorems 6 and 3). Ad piq ùñ piiq: If piq is true, since piq is equivalent with piiiq, it follows that there exists a disintegration tQ q…”

“…where the second equality follows from the assumption that the restriction of tR ψ u ψPR d is a disintegration of R over R Ψ consistent with Ψ, together with [8], Lemma 2.5 (i).…”