Compound counting processes in a periodic random environment

We briefly discuss the effects of a random environment, of repeated periodic nature, on the counting processes used in risk theory. A review of relevant results on such kind of periodic influence is given. An explicit formulation of counting processes in a periodic random environment is presented. Two renewal processes, known in reliability maintenance as minimal repair and replacement policy, are considered. Their properties are studied in the case where the generating random sequence has a distribution with periodic failure rate. Necessary and sufficient conditions for a non-stationary Poisson process to have periodic intensity are established. Representation of these processes as a finite sum of independent Poisson random variables and a limited-in-time Poisson process is shown. The transfer of these properties to compound processes is then discussed. Possible applications to risk theory are briefly considered.