We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of fO(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

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.