Asymptotics for Expectations of Hitting Times for Perturbed Semi-Markov Processes

Abstract

In this chapter, we apply asymptotic recurrent phase space reduction algorithms for regularly and singularly perturbed finite semi-Markov processes to an asymptotic analysis of expectations of hitting times and obtain theorems about convergence of expectations of hitting times for regularly and singularly perturbed finite semi-Markov processes. Special attention is paid to the conditions under which the expectations of hitting times, normalised by the functions used in the weak limit theorems for the distributions of the hitting times, converge to the first moments of the corresponding limiting distributions. This may not be the case for singularly perturbed semi-Markov processes. Such simultaneous convergence plays an important role in ergodic theorems for perturbed Markov type processes. This chapter includes three sections.