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May, 1995 Limits of First Passage Times to Rare Sets in Regenerative Processes
Paul Glasserman, Shing-Gang Kou
Ann. Appl. Probab. 5(2): 424-445 (May, 1995). DOI: 10.1214/aoap/1177004772

Abstract

We consider limits of first passage times to indexed families of nested sets in regenerative processes. The sets are exponentially rare, in the sense that the probability that the process reaches an indexed set in a cycle vanishes exponentially fast in the indexing parameter. Under appropriate formulations of this hypothesis, we prove strong laws, iterated logarithm laws and limits in distribution, both for the index of the rarest set reached in a cycle and for the time to reach a set. An interesting feature of the iterated logarithm laws is an asymmetry in the normalizations for the upper and lower limits. Our results apply to (possibly delayed) wide-sense regenerative processes, as well as those with independent cycles. We illustrate our results with queueing examples.

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Paul Glasserman. Shing-Gang Kou. "Limits of First Passage Times to Rare Sets in Regenerative Processes." Ann. Appl. Probab. 5 (2) 424 - 445, May, 1995. https://doi.org/10.1214/aoap/1177004772

Information

Published: May, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0830.60021
MathSciNet: MR1336877
Digital Object Identifier: 10.1214/aoap/1177004772

Subjects:
Primary: 60F20
Secondary: 60F15 , 60G70

Keywords: Extreme values , first passage times , Law of the iterated logarithm , limit theorems , Rare events , regenerative processes

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.5 • No. 2 • May, 1995
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