Laws of large numbers and functional central limit theorems for generalized semi-Markov processes

Because of the fundamental role played by generalized semi-Markov processes (GSMPs) in the modeling and analysis of complex discrete-event stochastic systems, it is important to understand the conditions under which a GSMP exhibits stable long-run behavior. To this end, we review existing work on strong laws of large numbers (SLLNs) and functional central limit theorems (FCLTs) for GSMPs; our discussion highlights the role played by the theory of both martingales and regenerative processes. We also sharpen previous limit theorems for finite-state irreducible GSMPs by establishing a SLLN and FCLT under the "natural" requirements of finite first (resp., second) moments on the clock-setting distribution functions. These moment conditions are comparable to the minimal conditions required in the setting of ordinary semi-Markov processes (SMPs). Corresponding discrete-time results for the underlying Markov chain of a GSMP are also provided. In contrast to the SMP setting, limit theorems for finite-state GSMPs require additional structural assumptions beyond irreducibility, due to the presence of multiple clocks. In our new limit theorems, the structural assumption takes the form of a "positive density" condition for specified clock-setting distributions. As part of our analysis, we show that finite moments for new clock readings imply finite moments for the od-regenerative cycles of both the GSMP and its underlying chain.