Maximal success durations for a semi-Markov process

Asymptotic distributional theorems are presented for the maximal sojourn duration on a semi-Markov chain and for the first passage time until a lengthy duration. Our analysis characterizes the limiting behavior of the longest “generalized success outcome” on a semi-Markov chain. The models include long runs on a single state, on a group of states, and on a set of transitions between states. Applications to success durations containing interruptions and to durations having infinite mean length are also presented.