, Volume 11, Issue 3, pp 235–259 | Cite as

Convex geometry of max-stable distributions

  • Ilya MolchanovEmail author


It is shown that max-stable random vectors in [0, ∞ ) d with unit Fréchet marginals are in one to one correspondence with convex sets K in [0, ∞ ) d called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function P ξ ≤ x of a max-stable random vector ξ with unit Fréchet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.


Copula Max-stable random vector Norm Cross-polytope Spectral measure Support function Zonoid 

AMS 2000 Subject Classifications

Primary—60G70 Secondary—52A21, 60D05 


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  1. Artstein, Z., Vitale, R.A.: A strong law of large numbers for random compact sets. Ann. Probab. 3, 879–882 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes. Theory and Applications. Wiley, Chichester (2004)zbMATHGoogle Scholar
  3. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, Berlin (1984)zbMATHGoogle Scholar
  4. Coles, S.G., Hefferman, J.E., Tawn, J.A.: Dependence measures for extreme value analyses. Extremes 2, 339–365 (1999)zbMATHCrossRefGoogle Scholar
  5. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B Stat. Methodol. 53, 377–392 (1991)zbMATHMathSciNetGoogle Scholar
  6. Davydov, Y., Molchanov, I., Zuyev, S.: Strictly stable distributions on convex cones. Electron. J. Probab. (2008, to appear)Google Scholar
  7. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  8. de Haan, L., de Ronde, J.: Sea and wind: multivariate extremes at work. Extremes 1, 7–45 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. de Haan, L., Ferreira, A.: Extreme Value Theory. An Introduction. Springer, New York (2006)zbMATHGoogle Scholar
  10. Einmahl, J.H.J., de Haan, L., Sinha, A.K.: Estimating the spectral measure of an extreme value distribution. Stoch. Process. Appl. 70, 143–171 (1997)zbMATHCrossRefGoogle Scholar
  11. Falk, M.: A representation of bivariate extreme value distributions via norms on ℝ2. Extremes 9, 63–68 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events, 2nd edn. Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  13. Falk, M., Reiss, R.-D.: On Pickands coordinates in arbitrary dimensions. J. Multivar. Anal. 92, 426–453 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  14. Firey, W.J.: Some means of convex bodies. Trans. Am. Math. Soc. 129, 181–217 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  15. Genest, C., Rivest, L.-P.: A characterization of Gumbel’s family of extreme value distributions. Stat. Probab. Lett. 8, 207–211 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  16. Giné, E., Hahn, M.G., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Relat. Fields 87, 139–165 (1990)zbMATHCrossRefGoogle Scholar
  17. Grünbaum, B.: Convex Polytopes. Wiley, London (1967)zbMATHGoogle Scholar
  18. Hall, P., Tajvidi, N.: Prediction regions for bivariate extreme events. Aust. N. Z. J. Stat. 46, 99–102 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  19. Hürlimann, W.: Hutchinson-Lai’s conjecture for bivariate extreme value copulas. Stat. Probab. Lett. 61, 191–198 (2003)zbMATHCrossRefGoogle Scholar
  20. Hüsler, J., Reiss, R.-D.: Maxima of normal random vectors: between independence and complete dependence. Stat. Probab. Lett. 7, 283–286 (1989)zbMATHCrossRefGoogle Scholar
  21. Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997)zbMATHGoogle Scholar
  22. Kotz, S., Nadarajah, S.: Extreme Value Distributions: Theory and Applications. Imperial College Press, London (2000)zbMATHGoogle Scholar
  23. Molchanov, I.: Theory of Random Sets. Springer, London (2005)zbMATHGoogle Scholar
  24. Nelsen, R.B.: Copulas and association. In: Dall’Aglio, G., Kotz, S., Salinetti, G. (eds.) Advances in Probability Distributions with Given Marginals, pp. 51–74. Kluwer, Dordrecht (1991)Google Scholar
  25. Nelsen, R.B.: An Introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer, New York (1999)Google Scholar
  26. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  27. Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  28. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer, Berlin (1987)zbMATHGoogle Scholar
  29. Ricker, W.: A new class of convex bodies. Contemp. Math. 9, 333–340 (1982)zbMATHMathSciNetGoogle Scholar
  30. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  31. Schlather, M., Tawn, J.A.: Inequalities for the extremal coefficients of multivariate extreme value distributions. Extremes 5, 87–102 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  32. Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90, 139–156 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  33. Schneider, R.: Convex Bodies. The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  34. Takahashi, R.: Asymptotic independence and perfect dependence of vector components of multivariate extreme statistics. Stat. Probab. Lett. 19, 19–26 (1994)zbMATHCrossRefGoogle Scholar
  35. Tawn, J.A.: Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  36. Thompson, A.C.: Minkowski Geometry. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  37. Vitale, R.A.: The Wills functional and Gaussian processes. Ann. Probab. 24, 2172–2178 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland

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