A regenerative stochastic process has the characteristic property that there exists an infinite sequence of random times at which the process probabilistically restarts. As discussed in Section 6.1, the essence of regeneration is that the evolution of the process between any two successive regeneration points is an independent probabilistic replica of the process in any other such “cycle.” Under mild regularity conditions, time-average limits for a regenerative process are well defined and finite, provided that the regenerative cycle length has finite mean. The value of a time-average limit is determined by the expected behavior of the process in a single regenerative cycle—a fact that has important implications for simulation analysis. Under some additional regularity conditions, the time-average limit can also be interpreted as a steady-state or limiting mean. Most of these results extend to the setting of “od-equilibrium” and “od-regenerative” processes. Such processes are similar to regenerative processes in that sample paths can be decomposed into identically distributed cycles, but differ in that adjacent cycles need not be independent.
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