The laws of iterated and triple logarithms for extreme values of regenerative processes

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 $\mathrm{limsup}$ and a law of the triple logarithm for the $\mathrm{liminf}$. 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.

Published at https://doi.org/10.15559/20-VMSTA147 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)