, Volume 13, Issue 1, pp 1–33 | Cite as

High-level dependence in time series models

  • Vicky Fasen
  • Claudia Klüppelberg
  • Martin Schlather
Open Access


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.


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 

AMS 2000 Subject Classifications

60G70 62G32 62M10 


  1. Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002a)MATHCrossRefMathSciNetGoogle Scholar
  2. Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115 (2002b)MATHCrossRefMathSciNetGoogle Scholar
  3. Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  5. Borkovec, M.: Extremal behavior of the autoregressive process with ARCH(1) errors. Stoch. Process. Appl. 85, 189–207 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 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)MATHCrossRefMathSciNetGoogle Scholar
  7. 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)MATHCrossRefMathSciNetGoogle Scholar
  8. Bougerol, P., Picard, N.: Stationarity of GARCH processes and some nonnegative time series. J. Econom. 52, 115–127 (1992a)MATHCrossRefMathSciNetGoogle Scholar
  9. Bougerol, P., Picard, N.: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20, 1714–1730 (1992b)MATHCrossRefMathSciNetGoogle Scholar
  10. Breiman, L.: On some limit theorems similar to the arc-sine law. Theory Probab. Appl. 10, 323–331 (1965)CrossRefMathSciNetGoogle Scholar
  11. Davis, R., Hsing, T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23, 879–917 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. Davis, R., Mikosch, T.: The extremogram: a correlogram for extreme events. Report, Department of Mathematics, University of Copenhagen, Denmark (2008)Google Scholar
  13. Davis, R., Resnick, S.: Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13, 179–195 (1985)MATHCrossRefMathSciNetGoogle Scholar
  14. 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)MATHCrossRefMathSciNetGoogle Scholar
  15. 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)MATHCrossRefGoogle Scholar
  16. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997)MATHGoogle Scholar
  17. Fasen, V.: Extremes of Lévy Driven MA Processes with Applications in Finance. Ph.D. thesis, Munich University of Technology (2004)Google Scholar
  18. Fasen, V.: Extremes of regularly varying mixed moving average processes. Adv. Appl. Probab. 37, 993–1014 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. Fasen, V.: Extremes of subexponential Lévy driven moving average processes. Stoch. Process. Appl. 116, 1066–1087 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. Fasen, V.: Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes. Bernoulli. (2009a)Google Scholar
  21. Fasen, V.: Extremes of mixed MA processes in the class of convolution equivalent distributions. Extremes. (2009b)Google Scholar
  22. Fasen, V., Klüppelberg, C.: Large insurance losses distributions. In: Everitt, B., Melnick, E. (eds.) Encyclopedia of Quantitative Risk Assessment. Wiley, New York (2008)Google Scholar
  23. Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991)MATHCrossRefMathSciNetGoogle Scholar
  24. 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)Google Scholar
  25. 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)Google Scholar
  26. Hult, H., Lindskog, F.: Extremal behavior for regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274 (2005)MATHCrossRefMathSciNetGoogle Scholar
  27. Hult, H., Samorodnitsky, G.: Tail probabilities for infinite series of regularly varying random vectors. Bernoulli. 14, 838–864 (2008)MATHCrossRefMathSciNetGoogle Scholar
  28. Jessen, A.H., Mikosch, T.: Regularly varying functions. Publications de l’Institute Mathé 80, 171–192 (2006)CrossRefMathSciNetGoogle Scholar
  29. Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MATHCrossRefMathSciNetGoogle Scholar
  30. 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)CrossRefGoogle Scholar
  31. 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)Google Scholar
  32. 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)MATHCrossRefMathSciNetGoogle Scholar
  33. Laurini, F., Tawn, J.A.: New estimators for the extremal index and other cluster characteristics. Extremes. 6, 189–211 (2003)MATHCrossRefMathSciNetGoogle Scholar
  34. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)Google Scholar
  35. Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. Roy. Statist. Soc. Ser. B 65, 521–543 (2003)MATHCrossRefMathSciNetGoogle Scholar
  36. 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)Google Scholar
  37. Mikosch, T., Stărică, C.: Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, 1427–1451 (2000)MATHCrossRefMathSciNetGoogle Scholar
  38. 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)Google Scholar
  39. Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econom. Theory 6, 318–334 (1990)CrossRefGoogle Scholar
  40. Ramos, A., Ledford, A.: A new class of models for bivariate joint tails. J. Roy. Statist. Soc. Ser. B 71 , 219–241 (2008)CrossRefGoogle Scholar
  41. Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)MATHGoogle Scholar
  42. Resnick, S.I.: Heavy-Tail Phenomena. Springer, New York (2007)MATHGoogle Scholar
  43. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  44. Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90, 139–156 (2003)MATHCrossRefMathSciNetGoogle Scholar
  45. Segers, J.: Multivariate regular variation of heavy-tailed Markov chains. Institut de statistique DP0703, available on arxiv.org. (2007)

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Vicky Fasen
    • 1
  • Claudia Klüppelberg
    • 1
  • Martin Schlather
    • 2
  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany
  2. 2.Centre for StatisticsUniversity of GöttingenGöttingenGermany

Personalised recommendations