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July 2007 Curve crossing for random walks reflected at their maximum
Ron Doney, Ross Maller
Ann. Probab. 35(4): 1351-1373 (July 2007). DOI: 10.1214/009117906000000953


Let Rn=max 0≤jnSjSn be a random walk Sn reflected in its maximum. Except in the trivial case when P(X≥0)=1, Rn will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of Rn above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of Sn is necessary for passage of Rn above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of Rn above linear and square root boundaries.


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Ron Doney. Ross Maller. "Curve crossing for random walks reflected at their maximum." Ann. Probab. 35 (4) 1351 - 1373, July 2007.


Published: July 2007
First available in Project Euclid: 8 June 2007

zbMATH: 1133.60022
MathSciNet: MR2330975
Digital Object Identifier: 10.1214/009117906000000953

Primary: 60F15 , 60G40 , 60J15 , 60K05
Secondary: 60F05 , 60G42

Keywords: power law boundaries , rate of growth , reflected process , Renewal theorems

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 4 • July 2007
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