, Volume 18, Issue 2, pp 191–212 | Cite as

Max-stable processes and the functional D-norm revisited

  • Stefan Aulbach
  • Michael Falk
  • Martin Hofmann
  • Maximilian ZottEmail author


Aulbach et al. (Extremes 16, 255283, 2013) introduced a max-domain of attraction approach for extreme value theory in C[0,1] based on functional distribution functions, which is more general than the approach based on weak convergence in de Haan and Lin (Ann. Probab. 29, 467483, 2001). We characterize this new approach by decomposing a process into its univariate margins and its copula process. In particular, those processes with a polynomial rate of convergence towards a max-stable process are considered. Furthermore we investigate the concept of differentiability in distribution of a max-stable processes.


Max-stable process D-norm Functional max-domain of attraction Copula process Generalized Pareto process δ-neighborhood of generalized Pareto process Derivative of D-norm Distributional differentiability 

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  1. Aulbach, S., Bayer, V., Falk, M.: A multivariate piecing-together approach with an application to operational loss data. Bernoulli 18, 455–475 (2012). doi: 10.3150/10-BEJ343 CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aulbach, S., Falk, M.: Local asymptotic normality in δ-neighborhoods of standard generalized Pareto processes. J. Statist. Plann. Inference 142, 1339–1347 (2012a). doi: 10.1016/j.jspi.2011.12.011 CrossRefzbMATHMathSciNetGoogle Scholar
  3. Aulbach, S., Falk, M.: Testing for a generalized Pareto process. Electron. J. Stat. 6, 1779–1802 (2012b). doi: 10.1214/12-EJS728 CrossRefzbMATHMathSciNetGoogle Scholar
  4. Aulbach, S., Falk, M., Hofmann, M.: On max-stable processes and the functional D-norm. Extremes 16, 255–283 (2013). doi: 10.1007/s10687-012-0160-3 CrossRefzbMATHMathSciNetGoogle Scholar
  5. Balkema, A.A., de Haan, L.: Residual life time at great age. Ann. Probab. 2, 792–804 (1974). doi: 10.1214/aop/1176996548 CrossRefzbMATHMathSciNetGoogle Scholar
  6. Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008). doi: 10.1214/08-AOAS159 CrossRefzbMATHMathSciNetGoogle Scholar
  7. Deheuvels, P.: Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes. Pub. Inst. Stat. Univ. Paris 23, 1–36 (1978)zbMATHGoogle Scholar
  8. Deheuvels, P.: Probabilistic aspects of multivariate extremes. In: de Oliveira, T.J. (ed.) Statistical Extremes and Applications, pp 117–130 (1984). D. Reidel DordrechtGoogle Scholar
  9. Dombry, C., Éyi-Minko, F.: Regular conditional distributions of continuous max-infinitely divisible random fields. Electron. J. Probab. 18, 1–21 (2013). doi: 10.1214/EJP.v18-1991 CrossRefMathSciNetGoogle Scholar
  10. Dombry, C., Éyi-Minko, F., Ribatet, M.: Conditional simulation of max-stable processes. Biometrika 100, 111–124 (2012). doi: 10.1093/biomet/ass067 CrossRefGoogle Scholar
  11. Dombry, C., Ribatet, M.: Functional regular variations, pareto processes and peaks over threshold type. Tech. Rep. (2013)Google Scholar
  12. Einmahl, J.H.J., Krajina, A., Segers, J.: An M-estimator for tail dependence in aribtrary dimensions. Ann. Statist. 40, 1764–1793 (2012). doi: 10.1214/12-AOS1023 CrossRefzbMATHMathSciNetGoogle Scholar
  13. Falk, M., Hüsler, J., Reiss, R.-D., 3rd: Laws of Small Numbers: Extremes and Rare Events. Springer, Basel (2011). doi: 10.1007/978-3-0348-0009-9 CrossRefGoogle Scholar
  14. Falk, M., Reiss, R.-D.: A characterization of the rate of convergence in bivariate extreme value models. Statist. Probab. Lett. 59, 341–351 (2002). doi: 10.1016/S0167-7152(02)00209-2 CrossRefzbMATHMathSciNetGoogle Scholar
  15. Ferreira, A., de Haan, L.: The generalized Pareto process; with a view towards application and simulation type Tech. Rep. To appear in Bernoulli. arXiv:math.PR1203.2551v2 (2012)
  16. Galambos, J., 1st: The Asymptotic Theory of Extreme Order Statistics Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1978)Google Scholar
  17. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204 (1984). doi: 10.1214/aop/1176993148 CrossRefzbMATHMathSciNetGoogle Scholar
  18. de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction Springer Series in Operations Research and Financial Engineering. Springer, New York (2006). See and for corrections and extensionsCrossRefGoogle Scholar
  19. de Haan, L., Lin, T.: On convergence toward an extreme value distribution in C[0,1]. Ann. Probab. 29, 467–483 (2001). doi: 10.1214/aop/1008956340 CrossRefzbMATHMathSciNetGoogle Scholar
  20. Pickands III, J.: Statistical inference using extreme order statistics. Ann. Statist. 3, 119–131 (1975). doi: 10.1214/aos/1176343003 CrossRefzbMATHMathSciNetGoogle Scholar
  21. Resnick, S.I., Roy, R.: Random usc functions, max-stable processes and continuous choice. Ann. Appl. Probab. 1, 267–292 (1991). doi: 10.1214/aoap/1177005937 CrossRefzbMATHMathSciNetGoogle Scholar
  22. Rootzén, H., Tajvidi, N.: Multivariate generalized Pareto distributions. Bernoulli 12, 917–930 (2006). doi: 10.3150/bj/1161614952 CrossRefzbMATHMathSciNetGoogle Scholar
  23. Wang, Y., Stoev, S.A.: Conditional sampling for spectrally discrete max-stable random fields. Adv. Appl. Probab. 43, 461–483 (2011) []CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Stefan Aulbach
    • 1
  • Michael Falk
    • 1
  • Martin Hofmann
    • 1
  • Maximilian Zott
    • 1
    Email author
  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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