Extreme sojourns for random walks and birth-and-death processes

Abstract

Let (X n) be a recurrent random walk on the nonnegative integers such that, at each stepX n increases or decreases by 1. The transition probabilities depend on the current state. For any starting point put visits to state n before the first visit to 0. Similarly, let (X t) be a recurrent birth-and-death process on the non-negative integers, and put L n = time spent in state n before the first passage time to 0. Exact asymptotic formulas are obtained for the tails of the distributions of M nand Ln , for under various conditions on the transition probabilities and the birth and death rates of the respective processes. For certain numbers (Bn ) associated with these processes, define the long term sojourns visits to n up to time [Bn ], and sojourn time in n up to time Bn . Under the additional hypothesis of ergodicity, it is shown what , suitably normalized, have limiting compound Poisson distributions, where the compounding distributions are the geometric and exponential, respectively