Abstract
We present several notions of high-level dependence for stochastic processes, which have appeared in the literature. We calculate such measures for discrete and continuous-time models, where we concentrate on time series with heavy-tailed marginals, where extremes are likely to occur in clusters. Such models include linear models and solutions to random recurrence equations; in particular, discrete and continuous-time moving average and (G)ARCH processes. To illustrate our results we present a small simulation study.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002a)
Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115 (2002b)
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Borkovec, M.: Extremal behavior of the autoregressive process with ARCH(1) errors. Stoch. Process. Appl. 85, 189–207 (2000)
Borkovec, M.: Asymptotic behaviour of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1) errors. Bernoulli 7, 847–872 (2001)
Borkovec, M., Klüppelberg, C.: The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Probab. 11, 1220–1241 (2001)
Bougerol, P., Picard, N.: Stationarity of GARCH processes and some nonnegative time series. J. Econom. 52, 115–127 (1992a)
Bougerol, P., Picard, N.: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20, 1714–1730 (1992b)
Breiman, L.: On some limit theorems similar to the arc-sine law. Theory Probab. Appl. 10, 323–331 (1965)
Davis, R., Hsing, T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23, 879–917 (1995)
Davis, R., Mikosch, T.: The extremogram: a correlogram for extreme events. Report, Department of Mathematics, University of Copenhagen, Denmark (2008)
Davis, R., Resnick, S.: Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13, 179–195 (1985)
Davis, R., Resnick, S.: Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68 (1988)
De Haan, L., Resnick, S.I., Rootzén, H., Vries, C.G.: Extremal behavior of solutions to a stochastic difference equation with applications to ARCH processes. Stoch. Process. Appl. 32, 213–224 (1989)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)
Fasen, V.: Extremes of Lévy Driven MA Processes with Applications in Finance. Ph.D. thesis, Munich University of Technology (2004)
Fasen, V.: Extremes of regularly varying mixed moving average processes. Adv. Appl. Probab. 37, 993–1014 (2005)
Fasen, V.: Extremes of subexponential Lévy driven moving average processes. Stoch. Process. Appl. 116, 1066–1087 (2006)
Fasen, V.: Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes. Bernoulli. (2009a)
Fasen, V.: Extremes of mixed MA processes in the class of convolution equivalent distributions. Extremes. (2009b)
Fasen, V., Klüppelberg, C.: Large insurance losses distributions. In: Everitt, B., Melnick, E. (eds.) Encyclopedia of Quantitative Risk Assessment. Wiley, New York (2008)
Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991)
Goldie, C.M., Klüppelberg, C.: Subexponential distributions. In: Adler, R.J., Feldman, R.E. (eds.) A Practical Guide to Heavy Tails: Statistical Techniques and Applications, pp. 435–459. Birkhäuser, Boston (1998)
Gomes, M.I., de Haan, L.D., Pestana, D.: Joint exceedances of the ARCH process. J. Appl. Probab. 41, 919–926. With correction in J. Appl. Prob. 43, 1206 (2006)
Hult, H., Lindskog, F.: Extremal behavior for regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274 (2005)
Hult, H., Samorodnitsky, G.: Tail probabilities for infinite series of regularly varying random vectors. Bernoulli. 14, 838–864 (2008)
Jessen, A.H., Mikosch, T.: Regularly varying functions. Publications de l’Institute Mathé 80, 171–192 (2006)
Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)
Klüppelberg, C., Lindner, A., Maller, R.: Continuous time volatility modelling: COGARCH versus Ornstein-Uhlenbeck models. In: Kabanov, Y., Liptser, R., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, pp. 393–419. Springer, Berlin (2006)
Klüppelberg, C.: Risk management with extreme value theory. In: Finkenstädt, B., Rootzén, H. (eds.) Extreme Values in Finance, Telecommunication and the Environment, pp. 101–168. Chapman & Hall/CRC, Boca Raton (2004)
Klüppelberg, C., Lindner, A., Maller, R.: A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour. J. Appl. Probab. 41, 601–622 (2004)
Laurini, F., Tawn, J.A.: New estimators for the extremal index and other cluster characteristics. Extremes. 6, 189–211 (2003)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)
Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. Roy. Statist. Soc. Ser. B 65, 521–543 (2003)
Mikosch, T.: Modeling dependence and tails of financial time series. In: Finkenstädt, B., Rootzén, H. (eds.) Extreme Values in Finance, Telecommunication and the Environment, pp. 185–286. Chapman & Hall/CRC, Boca Raton (2004)
Mikosch, T., Stărică, C.: Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, 1427–1451 (2000)
Naveau, P., Poncet, P., Cooley, D.: First-order variograms for extreme bivariate random vectors. Report, Laboratoire des Sciences du Climat et de l’Environnement, IPSL-CNRS, France (2008)
Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econom. Theory 6, 318–334 (1990)
Ramos, A., Ledford, A.: A new class of models for bivariate joint tails. J. Roy. Statist. Soc. Ser. B 71 , 219–241 (2008)
Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)
Resnick, S.I.: Heavy-Tail Phenomena. Springer, New York (2007)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90, 139–156 (2003)
Segers, J.: Multivariate regular variation of heavy-tailed Markov chains. Institut de statistique DP0703, available on arxiv.org. (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Parts of the paper were written while Vicky Fasen was visiting Cornell University. She thanks the Department of Operations Research and Industrial Engineering for their pleasant hospitality. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Fasen, V., Klüppelberg, C. & Schlather, M. High-level dependence in time series models. Extremes 13, 1–33 (2010). https://doi.org/10.1007/s10687-009-0084-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-009-0084-8
Keywords
- ARCH
- COGARCH
- Extreme cluster
- Extreme dependence measure
- Extremal index
- Extreme value theory
- GARCH
- Linear model
- Multivariate regular variation
- Nonlinear model
- Lévy-driven Ornstein–Uhlenbeck process
- Random recurrence equation