The laws of iterated and triple logarithms for extreme values of regenerative processes
Volume 7, Issue 1 (2020), pp. 61–78
Pub. online: 17 February 2020
Type: Research Article
Open Access
Open Access
Received
11 January 2020
11 January 2020
Revised
1 February 2020
1 February 2020
Accepted
1 February 2020
1 February 2020
Published
17 February 2020
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.
References
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). MR3104866. https://doi.org/10.1007/s11253-013-0716-7
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). MR256441. https://doi.org/10.1017/s0021900200026978
Asmussen, S.: Extreme value theory for queues via cycle maxima. Extremes 1(2), 137–168 (1998). MR1814621. https://doi.org/10.1023/A:1009970005784
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Encyclopedia of Mathematics and its Applications, vol. 27, p. 494. Cambridge University Press, Cambridge (1989). MR1015093
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). MR1952816. https://doi.org/10.1023/A:1020178327423
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). MR0232466
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). MR3927561
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events: for Insurance and Finance. Springer (2013) MR1458613. https://doi.org/10.1007/978-3-642-33483-2
Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II. Second edition, p. 669. John Wiley & Sons, Inc., New York-London-Sydney (1971). MR0270403
Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, p. 352. John Wiley & Sons, New York-Chichester-Brisbane (1978). Wiley Series in Probability and Mathematical Statistics. MR489334
Glasserman, P., Kou, S.-G.: Limits of first passage times to rare sets in regenerative processes. Ann. Appl. Probab. 5(2), 424–445 (1995). MR1336877
Gnedenko, B.V., Kovalenko, I.N.: Introduction to Queueing Theory, 2nd edn. Mathematical Modeling, vol. 5, p. 314. Birkhäuser Boston, Inc., Boston, MA (1989). https://doi.org/10.1007/978-1-4615-9826-8. Translated from the second Russian edition by Samuel Kotz. MR1035707
Iglehart, D.L.: Extreme values in the $GI/G/1$ queue. Ann. Math. Stat. 43, 627–635 (1972). MR305498. https://doi.org/10.1214/aoms/1177692642
Karlin, S.: A First Course in Stochastic Processes, p. 502. Academic Press, New York-London (1966). MR0208657
Karlin, S., McGregor, J.: The classification of birth and death processes. Trans. Am. Math. Soc. 86, 366–400 (1957). MR94854. https://doi.org/10.2307/1993021
Klass, M.: The minimal growth rate of partial maxima. Ann. Probab. 12(2), 380–389 (1984) MR0735844
Klass, M.: The Robbins–Siegmund series criterion for partial maxima. Ann. Probab. 13(4), 1369–1370 (1985) MR0806233
Robbins, H., Siegmund, D.: On the law of the iterated logarithm for maxima and minima. In: Proc. 6th Berkeley Symp. Math. Stat. Prob, vol. 3, pp. 51–70 (1972) MR0400364
Serfozo, R.F.: Extreme values of birth and death processes and queues. Stoch. Process. Appl. 27(2), 291–306 (1988). MR931033. https://doi.org/10.1016/0304-4149(87)90043-3
Smith, W.L.: Renewal theory and its ramifications. J. R. Stat. Soc. Ser. B 20, 243–302 (1958). MR99090
Zakusilo, O.K., Matsak, I.K.: On the extreme values of some regenerative processes. Teor. Imovir. Mat. Stat. 97, 58–71 (2017). MR3745999. https://doi.org/10.1090/tpms/1048
