Conditioned Limit Theorems for Some Null Recurrent Markov Processes

Abstract

Let $\{v_k, k \geqq 0\}$ be a discrete time Markov process with state space $E \subset (-\infty, \infty)$ and let $S$ be a proper subset of $E$. In several applications it is of interest to know the behavior of the system after a large number of steps, given that the process has not entered $S$. In this paper we show that under some mild restrictions there is a functional limit theorem for the conditioned sequence if there was one for the original sequence. As applications we obtain results for branching processes, random walks, and the M/G/1 queue which complete or extend the work of previous authors. In addition we consider the convergence of conditioned birth and death processes and obtain results which are complete except in the case that 0 is an absorbing boundary.