Aggregated Markov processes incorporating time interval omission

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed .open' and .closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.