Limit laws for random vectors with an extreme component

Annals of Applied Probability

Volumn 17, Issue 2, 2007, Pages 537-571

  Heffernan, Janet E ^a,c   Resnick, Sidney I ^b,d  

^a LANCASTER UNIVERSITY   (United Kingdom)

^b Cornell University ^*

^c LANCASTER UNIVERSITY   (United Kingdom)

^d School of Operations Research and Industrial Engineering   (United States)

Author keywords

Asymptotic independence; Coefficient of tail dependence; Conditional models; Heavy tails; Hidden regular variation; Regular variation

Indexed keywords

EID: 40249096859     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051606000000835     Document Type: Article

References (40)

1. ABDOUS, B., FOUGÈRES, A.-L. and GHOUDI, K. (2005). Extreme behaviour for bivariate elliptical distributions. Canad. J. Statist. 33 317-334. MR2193978
2. BALKEMA, A.A. (1973). Monotone Transformations and Limit Laws. Mathematisch Centrum, Amsterdam. MR0334307
3. BASRAK, B., DAVIS, R. and MIKOSCH, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908-920. MR1925445
4. BINGHAM, N., GOLDIE, C. and TEUGELS, J. (1987). Regular Variation. Cambridge Univ. Press. MR0898871
5. COLES, S. G. and TAWN, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377-392. MR1108334
6. 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.
7. COLES, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London. MR1932132
8. DAVISON, A. C. and SMITH, R. L. (1990). Models for exceedances overhigh thresholds (with discussion). J. Roy. Statist. Soc. Ser. B 52 393-442. MR1086795
9. EMBRECHTS, P., KLUPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extreme Events for Insurance and Finance. Springer, Berlin. MR1458613
10. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York.
11. GELUK, J. L. and DE HAAN, L. (1987). Regular Variation, Extensions and Tauberian Theorems. Stichting Mathematisch Centrum, Amsterdam. MR0906871
12. DE HAAN, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Math. Centrum, Amsterdam. MR0286156
13. DE HAAN, L. (1974). Equivalence classes of regularly varying functions. Stochastic Process. Appl. 2 243-259. MR0365664
14. DE HAAN, L. (1976). An Abel-Tauber theorem for Laplace transforms. J. London Math. Soc. (2) 13 537-542. MR0407542
15. 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. LSI, Hague, Netherlands. MR0886266
16. DE HAAN, L. and DE RONDE, J. (1998). Sea and wind: Multivariate extremes at work. Extremes 17-46. MR1652944
17. DE HAAN, L. and FERREIRA, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. MR2234156
18. DE HAAN, L. and RESNICK, S. I. (1979). Conjugate π-variation and process inversion. Ann. Probab. 7 1028-1035. MR0548896
19. DE HAAN, L. and RESNICK, S. (1977). Limit theory for multivariate sample extremes. Z Wahrsch. Verw. Gebiete 40 317-337. MR0478290
20. HEFFERNAN, J. and RESNICK, S. (2005). Hidden regular variation and the rank transform. Adv. in Appl. Probab. 37 393-414. MR2144559
21. 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. MR2088289
22. KALLENBERG, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin. MR0818219
23. KURATOWSKI, K. (1966). Topology. I. New edition, revised and augmented. Translated from the French, by J. Jaworowski. Academic Press, New York. MR0217751
24. LEDFORD, A. and TAWN, J. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169-187. MR1399163
25. LEDFORD, A. and TAWN, J. (1997). Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B 59 475-499. MR1440592
26. LEDFORD, A. and TAWN, J. (1998). Concomitant tail behaviour for extremes. Adv. in Appl. Probab. 30 197-215. MR1618837
27. MAULIK, K. and RESNICK, S. (2005). Characterizations and examples of hidden regular variation. Extremes 7 31-67. MR2201191
28. MAULIK, K., RESNICK, S. and ROOTZÉN, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Probab. 39 671-699. MR1938164
29. NEVEU, J. (1977). Processus ponctuels. École d'Été de Probabilités de Saint-Flour VI-1976. Lecture Notes in Math. 598 249-445. Springer, Berlin. MR0474493
30. PICKANDS, J. (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statististical Institute 2 859-878. MR0820979
31. REISS, R.-D. and THOMAS, M. (2001). Statistical Analysis of Extreme Values, 2nd ed. Birkhäuser, Basel. MR1819648
32. RESNICK, S. (2006). Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
33. RESNICK, S. (1973). Limit laws for record values. Stochastic Process. Appl. 1 67-82. MR0362454
34. RESNICK, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York. MR0900810
35. RESNICK, S. (1999). A Probability Path. Birkhäuser, Boston. MR1664717
36. RESNICK, S. (2002). Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5 303-336. MR2002121
37. 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.
38. J. Appl. Probab. 41A. Papers in honour of C. C. Heyde. MR2057574
39. SENETA, E. (1976). Regularly Varying Functions. Springer, New York. MR0453936
40. 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. MR1041763