Applications of simple point process methods to superpositions of aggregated stationary processes

For a stationary simple point process, a fundamental relationship is that its intensity is equal to the reciprocal of the mean inter-point time. The present paper is concerned with applications of this result to aggregated processes, that is to processes whose states are classes of states in some partition of the state space of an underlying process. It is shown that the intensity relationship and extensions of it provide a unified approach to moments of class sojourn times for aggregated processes and superpositions of independent and identically distributed aggregated processes, and often a simple method for calculating such moments. Extensions of the method to moments of sojourn-times for superpositions of dependent aggregated processes are also considered.