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October, 1978 Conditioned Limit Theorems for Some Null Recurrent Markov Processes
Richard Durrett
Ann. Probab. 6(5): 798-828 (October, 1978). DOI: 10.1214/aop/1176995430


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.


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Richard Durrett. "Conditioned Limit Theorems for Some Null Recurrent Markov Processes." Ann. Probab. 6 (5) 798 - 828, October, 1978.


Published: October, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0398.60023
MathSciNet: MR503953
Digital Object Identifier: 10.1214/aop/1176995430

Primary: 60F05
Secondary: 60J15 , 60J80 , 60K25

Keywords: $M/G/1$ queue , birth and death processes , branching processes , conditioned limit theorems , Diffusions , Functional limit theorems , Random walks

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 5 • October, 1978
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