On Functional Central Limit Theorems for Semi-Markov and Related Processes

The semi-Markov process (SMP) has long been used as a model for the underlying process of a discrete-event stochastic system. Important refinements of this model include the continuous-time Markov chain (CTMC) and important extensions include the generalized semi-Markov process (GSMP). Functional central limit theorems (FCLTS) give basic conditions under which these various processes exhibit stable long-run behavior, as well as providing approximations for cumulative-reward distributions and confidence intervals for statistical estimators. We give FCLTS for finite-state CTMCS, SMPS, and GSMPS under minimal conditions that involve irreducibility and finite second moments on the "holding time" distributions. We consider both continuous and lump-sum rewards; our emphasis is on the use of martingale theory and on the explicit computation, when possible, of the variance constant in the FCLT.