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August, 1981 Growth of Random Walks Conditioned to Stay Positive
Grant A. Ritter
Ann. Probab. 9(4): 699-704 (August, 1981). DOI: 10.1214/aop/1176994378


For random walk $S_k = \sum^k_{i = 1} \xi_i$ let $T$ be the hitting time of the lower half plane. By conditioning the process $\{S_k\}^n_{k = 1}$ relative to $\lbrack T > n\rbrack$ we create a "random walk conditioned to stay positive." Sample paths of such processes tend to grow rather quickly. In studying this growth we find that except for a set of probability $\epsilon$ all such sample paths have as lower bounds any sequence of the form $\{\delta k^\eta\}^n_{k = 1}$ where $\eta \in (0, 1/2)$ and $\delta < \delta(\epsilon, \eta)$. Applications of this result to sample path behavior of a random walk as it approaches or leaves the lowest of its first $n$ values are also given.


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Grant A. Ritter. "Growth of Random Walks Conditioned to Stay Positive." Ann. Probab. 9 (4) 699 - 704, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60041
MathSciNet: MR624698
Digital Object Identifier: 10.1214/aop/1176994378

Primary: 60G17
Secondary: 60J15

Keywords: Random walk , random walk conditioned to stay positive

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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