Rarity and Exponentiality

Abstract

If a system is modeled by a finite Markov chain which is ergodic, the passage time from some specified initial distribution over the state space to a subset B of the state space visited infrequently is often exponentially distributed to good approximation. The chapter is devoted to the limit theorems surrounding such behavior for processes and the characterization of thè circumstances under which exponentiality is present. In the absence of certain “jitter,” i.e., clustering of the entry epochs into the good set G, the time to failure from the perfect state, the quasi-stationary exit time, the ergodic exit time and the sojourn time on the good set then have a common asymptotic exponential distribution and common expectations. For engineering purposes, it is essential to quantify departure from exponentiality via error bounds. When one is dealing with time-reversible chains e.g., systems with independent Markov components, the complete monotonicity present permits such quantification and the error bounds needed.