Abstract
In practice it is important to evaluate the impact of clusters of extreme observations caused by the dependence in time series. The clusters contain consecutive exceedances of time series over a threshold separated by return intervals with consecutive non-exceedances. We derive asymptotically equal distributions of the number of inter-arrival times between events of interest arising both between two consecutive exceedances of a stationary process \(\{R_n:n\ge 1\}\) and between two consecutive non-exceedances. It is found that the distributions are geometric like and corrupted by the extremal index. It is derived that the limit distribution tail of the duration of clusters that is defined as a sum of the random number of the weakly dependent regularly varying inter-arrival times with tail index \(0<\alpha <2\) is bounded by the tail of stable distribution. The inferences are valid when the threshold is taken as a sufficiently high quantile of the underlying process \(\{R_{n}\}\).
Similar content being viewed by others
References
Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of methods for estimating the extremal index. Extremes 3(1), 5–38 (2000)
Bartkiewicz, K., Jakubowski, A., Mikosch, T., Wintenberger, O.: Stable limits for sums of dependent infinite variance random variables. Probab. Theory Relat. Fields (2010). doi:10.1007/s00440-010-0276-9
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)
Basrak, B., Krizmanić, D., Segers, J.: A functional limit theorem for partial sums of dependent random variables with infinite variance. http://arxiv.org/abs/1001.1345. Accessed 8 Jan 2010 (2010)
Billingsley, P.: Convergence of Probability Measures, 2nd ed. Wiley, New York (1999)
Beirlant, J., Goegebeur, Y., Teugels, J., Segers, J.: Statistics of Extremes: Theory and Applications, vol. 504. Wiley, Chichester, West Sussex (2004)
Bradley, R.C.: A central limit theorem for stationary ρ- mixing sequences with infinite variance. Ann. Probab. 16, 313–332 (1988)
Chernick, M.R., Hsing, T., McCormick, W.P.: Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835–850 (1991)
D’Auria, B., Resnick, S.I.: The influence of dependence on data network models of burstiness. Adv. Appl. Prob. 40(1), 60–94 (2008)
Embrechts, P., Klűppelberg, C., Mikosch, T.: Modeling Extremal Events for Finance and Insurance. Springer, Berlin (1997)
Eichner, J.F., Kantelhardt, J.W., Bunde, A., Havlin, S.: Statistics of return intervals in long-term correlated records. Phys. Rev. E 75, 011128 (2007)
Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Stat. Soc. B 65, 545–556 (2003)
Hsing, T., Huesler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Prob. Theory Relat. Fields 78, 97–112 (1988)
Jessen, A.H., Mikosch, T.: Regularly varying functions. Publications de lInstitut Mathématique Nouvelle série, tome 80(94), 171192 (2006)
Krizmanić, D.: Functional limit theorems for weakly dependent regularly varying time series. PhD thesis, Univ. Zagreb. Available at http://www.math.uniri.hr/dkrizmanic/DKthesis.pdf (2010)
Leadbetter, M.R., Lingren, G., Rootzén, H.: Extremes and Related Properties of Random Sequence and Processes, ch. 3. Springer, New York (1983)
Leadbetter, M.R., Nandagopalan, L.: On exceedance point processes for stationary sequences under mild oscillation restrictions. Lect. Notes Statist. 51, 69–80 (1989)
Markovich, N.M., Undheim, A., Emstad, P.J.: Classification of slice-based VBR video traffic. Comput. Netw. 53, 1137–1153 (2009)
Markovich, N.M., Kilpi, J.: Bivariate statistical analysis of TCP-flow sizes and durations. Ann. Oper. Res. 170(1), 199–216 (2009)
Markovich, N.M.: Modeling of dependence in a peer-to-peer video application. In: Proceedings of the International Wireless Communications and Mobile Computing Conference (IWCMC 2010), pp. 316–320. Caen, France (2010)
Markovich, N.M., Krieger, U.R.: Statistical analysis and modeling of peer-to-peer multimedia traffic. In: Kouvatsos, D. (ed.) Next Generation Internet: Performance Evaluation and Applications, LNCS 5233, pp. 37–69. Springer, Heidelberg (2011)
Merlevède, F., Peligrad, M., Utev, S.: Recent advances in invariance principles for stationary sequences. Probab. Surv. 3, 1–36 (2006). doi:10.1214/154957806100000202
Nolan, J.P.: Stable distributions: models for heavy tailed data. In: Progress, ch. 1, Birkhauser, Boston. http://academic2.american.edu/jpnolan/stable/chap1.pdf (2011)
O’Brien, G.L.: Extreme values for stationary and Markov sequences. Ann. Probab. 15(1), 281–291 (1987)
Resnick, S.I.: Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer, New York (2006)
Robert, C.Y., Segers, J.: Tails of random sums of a heavy-tailed number of light-tailed terms. Insur. Math. Econ. 43, 8592 (2008)
Robinson, M.E., Tawn, J.A.: Extremal analysis of processes sampled at different frequences. J. R. Stat. Soc. Ser. B 62(1), 117135 (2000). doi:10.1111/1467-9868.00223
Santhanam, M.S., Kantz, H.: Return interval distribution of extreme events and long term memory. Phys. Rev. E 78, 05113 (2008)
Tyran-Kamińska, M.: Convergence to Levy stable processes under some weak dependence conditions. Stoch. Process. Appl. 120, 1629–1650 (2010)
Whitt, W.: Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and their Application to Queues. Springer, New York (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Markovich, N.M. Modeling clusters of extreme values. Extremes 17, 97–125 (2014). https://doi.org/10.1007/s10687-013-0176-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-013-0176-3