Marynych, Alexander; Matsak, Ivan The laws of iterated and triple logarithms for extreme values of regenerative processes. (English) Zbl 1435.60041 Mod. Stoch., Theory Appl. 7, No. 1, 61-78 (2020). Summary: 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 P. Glasserman and S.-G. Kou [Ann. Appl. Probab. 5, No. 2, 424–445 (1995; Zbl 0830.60021)]. We apply our results to several queuing systems and a birth and death process. MSC: 60G70 Extreme value theory; extremal stochastic processes 60F15 Strong limit theorems 60K25 Queueing theory (aspects of probability theory) Keywords:extreme values; regenerative processes; queuing systems Citations:Zbl 0830.60021 PDFBibTeX XMLCite \textit{A. Marynych} and \textit{I. Matsak}, Mod. Stoch., Theory Appl. 7, No. 1, 61--78 (2020; Zbl 1435.60041) Full Text: DOI arXiv References: [1] Akbash, K. S.; Matsak, I. K., One improvement of the law of the iterated logarithm for the maximum scheme, Ukr. Math. J., 64, 8, 1290-1296 (2013) · Zbl 1274.60085 [2] Anderson, C. W., Extreme value theory for a class of discrete distributions with applications to some stochastic processes, J. Appl. Probab., 7, 99-113 (1970) · Zbl 0192.54202 [3] Asmussen, S., Extreme value theory for queues via cycle maxima, Extremes, 1, 2, 137-168 (1998) · Zbl 0926.60072 [4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation, 27, 494 (1989), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0667.26003 [5] Buldygin, V. V.; Klesov, O. I.; Steinebach, J. G., Properties of a subclass of Avakumović functions and their generalized inverses, Ukr. Math. J., 54, 2, 149-169 (2002) · Zbl 1006.60087 [6] Cohen, J. W., Extreme value distribution for the \(M/G/1\) and the \(G/M/1\) queueing systems, Ann. Inst. Henri Poincaré Sect. B (N.S.), 4, 83-98 (1968) · Zbl 0162.49302 [7] Dovgaĭ, B. V.; Matsak, I. K., Asymptotic behavior of the extreme values of the queue length in \(M/M/m\) queuing systems, Kibern. Sist. Anal., 55, 2, 171-179 (2019) [8] Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events: for Insurance and Finance (2013), Springer [9] Feller, W., An Introduction to Probability Theory and Its Applications. Vol. II, 669 (1971), John Wiley & Sons, Inc.: John Wiley & Sons, Inc., New York-London-Sydney · Zbl 0219.60003 [10] Galambos, J., The Asymptotic Theory of Extreme Order Statistics, 352 (1978), John Wiley & Sons: John Wiley & Sons, New York-Chichester-Brisbane [11] Glasserman, P.; Kou, S.-G., Limits of first passage times to rare sets in regenerative processes, Ann. Appl. Probab., 5, 2, 424-445 (1995) · Zbl 0830.60021 [12] Gnedenko, B. V.; Kovalenko, I. N., Introduction to Queueing Theory, 5, 314 (1989), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA · Zbl 0744.60111 [13] Iglehart, D. L., Extreme values in the \(GI/G/1\) queue, Ann. Math. Stat., 43, 627-635 (1972) · Zbl 0238.60072 [14] Karlin, S., A First Course in Stochastic Processes, 502 (1966), Academic Press: Academic Press, New York-London [15] Karlin, S.; McGregor, J., The classification of birth and death processes, Trans. Am. Math. Soc., 86, 366-400 (1957) · Zbl 0091.13802 [16] Klass, M., The minimal growth rate of partial maxima, Ann. Probab., 12, 2, 380-389 (1984) · Zbl 0536.60038 [17] Klass, M., The Robbins-Siegmund series criterion for partial maxima, Ann. Probab., 13, 4, 1369-1370 (1985) · Zbl 0576.60023 [18] Matsak, I. K., Limit theorem for extreme values of discrete random variables and its application, Teor. Imovir. Mat. Stat., 101, 189-202 (2019) [19] Robbins, H.; Siegmund, D., Proc. 6th Berkeley Symp. Math. Stat. Prob, 3, On the law of the iterated logarithm for maxima and minima, 51-70 (1972) · Zbl 0281.60027 [20] Serfozo, R. F., Extreme values of birth and death processes and queues, Stoch. Process. Appl., 27, 2, 291-306 (1988) · Zbl 0637.60098 [21] Smith, W. L., Renewal theory and its ramifications, J. R. Stat. Soc. Ser. B, 20, 243-302 (1958) · Zbl 0091.30101 [22] Zakusilo, O. K.; Matsak, I. K., On the extreme values of some regenerative processes, Teor. Imovir. Mat. Stat., 97, 58-71 (2017) · Zbl 1409.60136 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.