The Number of Visits to a Subset of the State Space by an Irreducible Semi-Markov Process During a Finite Time Interval: Moment Results

Abstract

The present and the next chapter are devoted to the examination of the variable ${M_{{A_1}}}(t)$ from Chapter 5, now under the semi-Markov assumption. The notation and the assumptions of Chapter 8 apply. We assume here that Y is irreducible and that the state space is partitioned into two subsets, i.e., S = A₁ ∪ A₂. Furthermore, we require that the holding time cumulative distribution functions satisfy ${F_{s,s'}}(0) < 1$ for all s, s’ ∈ S, s ≠ s’. As we want to concentrate here on the moments of ${M_{{A_1}}}(t)$, the following elementary lemma will prove useful.