Stationary generalized regenerative processes

Based on the so-called «inversion formula» from the theory of stochastic point processes, the concept of stationary and synchroneous version of a generalized regenerative process is introduced. Such processes are generated by given strictly stationary sequences of regeneration cycles without any independence assumptions in general. An ergodic theorem and a formula are proved which expresses the stationary distribution of the process as the time average of the process over a regeneration cycle. Some well-known classes of stochastic processes are special cases of the considered model when the cycles form a Markov chain.