Asymptotics of Hitting Times for Perturbed Semi-Markov Processes

This chapter plays a key role in Part II. Here we apply asymptotic recurrent phase space reduction algorithms for regularly and singularly perturbed finite semi-Markov processes to an asymptotic analysis of distributions of hitting times. It is important that the hitting times are asymptotically invariant in distribution with respect to the proposed procedures for reducing the phase space. We formulate conditions that ensure that the basic perturbation conditions imposed on the original semi-Markov processes are also satisfied for semi-Markov processes with reduced phase spaces. We also give recurrent formulas for recalculating normalisation functions, limiting distributions and expectations appearing in the corresponding perturbation conditions for semi-Markov processes with reduced phase spaces. Acting in this way, we recursively reduce the asymptotic analysis to the case, where the hitting time coincides with the first transition time for the corresponding reduced semi-Markov processes and obtain theorems on weak convergence of hitting times for regularly and singularly perturbed finite semi-Markov processes. This chapter includes three sections.