The Annals of Applied Probability

Limit laws for random vectors with an extreme component

Janet E. Heffernan and Sidney I. Resnick

Full-text: Open access


Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence structures when marginal distributions are heavy tailed. Multivariate regular variation provides a rich description of extremal dependence in the case of asymptotic dependence, but fails to distinguish between exact independence and asymptotic independence. Hidden regular variation addresses this problem by requiring components of the random vector to be simultaneously large but on a smaller scale than the scale for the marginal distributions. In doing so, hidden regular variation typically restricts attention to that part of the probability space where all variables are simultaneously large. However, since under asymptotic independence the largest values do not occur in the same observation, the region where variables are simultaneously large may not be of primary interest. A different philosophy was offered in the paper of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497–546] which allows examination of distributional tails other than the joint tail. This approach used an asymptotic argument which conditions on one component of the random vector and finds the limiting conditional distribution of the remaining components as the conditioning variable becomes large. In this paper, we provide a thorough mathematical examination of the limiting arguments building on the orientation of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497–546]. We examine the conditions required for the assumptions made by the conditioning approach to hold, and highlight simililarities and differences between the new and established methods.

Article information

Ann. Appl. Probab., Volume 17, Number 2 (2007), 537-571.

First available in Project Euclid: 19 March 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62G32: Statistics of extreme values; tail inference

Conditional models heavy tails regular variation coefficient of tail dependence hidden regular variation asymptotic independence


Heffernan, Janet E.; Resnick, Sidney I. Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17 (2007), no. 2, 537--571. doi:10.1214/105051606000000835.

Export citation


  • Abdous, B., Fougères, A.-L. and Ghoudi, K. (2005). Extreme behaviour for bivariate elliptical distributions. Canad. J. Statist. 33 317–334.
  • Balkema, A. A. (1973). Monotone Transformations and Limit Laws. Mathematisch Centrum, Amsterdam.
  • Basrak, B., Davis, R. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908–920.
  • Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge Univ. Press.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377–392.
  • Coles, S. G. and Tawn, J. A. (1994). Statistical methods for multivariate extremes: An application to structural design (with discussion). J. R. Stat. Soc. Ser. C 43 1–48.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  • Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds (with discussion). J. Roy. Statist. Soc. Ser. B 52 393–442.
  • Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997). Modelling Extreme Events for Insurance and Finance. Springer, Berlin.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York.
  • Geluk, J. L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems. Stichting Mathematisch Centrum, Amsterdam.
  • de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Math. Centrum, Amsterdam.
  • de Haan, L. (1974). Equivalence classes of regularly varying functions. Stochastic Process. Appl. 2 243–259.
  • de Haan, L. (1976). An Abel–Tauber theorem for Laplace transforms. J. London Math. Soc. (2) 13 537–542.
  • de Haan, L. (1985). Extremes in higher dimensions: The model and some statistics. In Proceedings of the 45th Session of the International Statistical Institute 4 (Amsterdam, 1985) 185–192. Bull. Inst. Internat. Statist. 51. ISI, Hague, Netherlands.
  • de Haan, L. and de Ronde, J. (1998). Sea and wind: Multivariate extremes at work. Extremes 1 7–46.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • de Haan, L. and Resnick, S. I. (1979). Conjugate $\pi$-variation and process inversion. Ann. Probab. 7 1028–1035.
  • de Haan, L. and Resnick, S. (1977). Limit theory for multivariate sample extremes. Z. Wahrsch. Verw. Gebiete 40 317–337.
  • Heffernan, J. and Resnick, S. (2005). Hidden regular variation and the rank transform. Adv. in Appl. Probab. 37 393–414.
  • Heffernan, J. and Tawn. J. (2004). A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 66 497–546.
  • Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin.
  • Kuratowski, K. (1966). Topology. I. New edition, revised and augmented. Translated from the French by J. Jaworowski. Academic Press, New York.
  • Ledford, A. and Tawn, J. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • Ledford, A. and Tawn, J. (1997). Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B 59 475–499.
  • Ledford, A. and Tawn, J. (1998). Concomitant tail behaviour for extremes. Adv. in Appl. Probab. 30 197–215.
  • Maulik, K. and Resnick, S. (2005). Characterizations and examples of hidden regular variation. Extremes 7 31–67.
  • Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Probab. 39 671–699.
  • Neveu, J. (1977). Processus ponctuels. École d'Été de Probabilités de Saint-Flour VI–-1976. Lecture Notes in Math. 598 249–445. Springer, Berlin.
  • Pickands, J. (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statististical Institute 2 859–878.
  • Reiss, R.-D. and Thomas, M. (2001). Statistical Analysis of Extreme Values, 2nd ed. Birkhäuser, Basel.
  • Resnick, S. (2006). Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • Resnick, S. (1973). Limit laws for record values. Stochastic Process. Appl. 1 67–82.
  • Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Resnick, S. (1999). A Probability Path. Birkhäuser, Boston.
  • Resnick, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5 303–336.
  • Resnick, S. (2004). On the foundations of multivariate heavy-tail analysis. In Stochastic Methods and Their Applications (J. Gani and E. Seneta, eds.) 191–212. J. Appl. Probab. 41A. Papers in honour of C. C. Heyde.
  • Seneta, E. (1976). Regularly Varying Functions. Springer, New York.
  • Smith, R. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground level ozone (with discussion). Statist. Sci. 4 367–393.