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April, 1977 A Ratio Limit Theorem for Subterminal Times
Samuel D. Oman
Ann. Probab. 5(2): 262-277 (April, 1977). DOI: 10.1214/aop/1176995850


Consider a recurrent random walk $\{X_n\}$ with state space $S \subseteqq R^d (d \leqq 2)$. A stopping time $T$ is called subterminal if it satisfies a technical condition which essentially states that it is the first time a path possesses some property which does not depend on how long the process has been running. Suppose $T$ is a subterminal time which can occur only when $\{X_n\}$ is in a bounded set; then under an additional assumption a ratio limit theorem (as $n \rightarrow \infty$) is obtained for $P(T > n \mid X_0 = x) (x\in S)$. The theorem applies in particular to the hitting time of a bounded set with nonempty interior in the general case, and to the hitting time of a bounded set with nonzero Haar measure in the nonsingular case.


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Samuel D. Oman. "A Ratio Limit Theorem for Subterminal Times." Ann. Probab. 5 (2) 262 - 277, April, 1977.


Published: April, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0368.60079
MathSciNet: MR436332
Digital Object Identifier: 10.1214/aop/1176995850

Primary: 60J15
Secondary: 60F99

Keywords: hitting times , Random walk , ratio limit theorem , terminal times

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 2 • April, 1977
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