Aggregated Markov processes with negative exponential time interval omission

A time reversible, continuous time Markov chain on a finite state space is considered. 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. The situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean is considered. It is shown that the probability density function of observed open sojourns takes the form , where n is the size of the state space. A thorough asymptotic analysis of as tends to infinity is presented. The relevance of the present results to the modelling of single channel records is discussed. The theory is illustrated with a numerical example.