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The laws of iterated and triple logarithms for extreme values of regenerative processes
Volume 7, Issue 1 (2020), pp. 61–78
Alexander Marynych   Ivan Matsak  

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https://doi.org/10.15559/20-VMSTA147
Pub. online: 17 February 2020      Type: Research Article      Open accessOpen Access

Received
11 January 2020
Revised
1 February 2020
Accepted
1 February 2020
Published
17 February 2020

Abstract

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the lim sup and a law of the triple logarithm for the lim inf. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424–445]. We apply our results to several queuing systems and a birth and death process.

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Keywords
Extreme values regenerative processes queuing systems

MSC2010
60G70 60F15 60K25

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