State aggregation and discrete-state Markov chains embedded in point processes

One result of both theoretic and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a subsequence of the Poisson process. The authors extend this result by showing that on the embedded Poisson process a Markov chain can be defined with a discrete state that characterizes the stage of the interarrival times. This implies that embedded Markov chains can be constructed with countable state spaces for the state processes of many practical systems that can be modeled by point processes