Aggregated Markov processes with negative exponential time interval omission

Abstract

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, wherenis the size of the state space. We present a thorough asymptotic analysis offO(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.