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Asymptotic Dependence for Light-Tailed Homothetic Densities

Part of: Stochastic processes Distribution theory

Published online by Cambridge University Press:  04 January 2016

Guus Balkema  and
Natalia Nolde
Guus Balkema*
Affiliation:
University of Amsterdam
Natalia Nolde*
Affiliation:
University of British Columbia
*
∗ Postal address: Department of Mathematics, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands. Email address: [email protected]
∗∗ Postal address: Department of Statistics, University of British Columbia, 6356 Agricultural Road, Vancouver, BC V6T 1Z2, Canada. Email address: [email protected]
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Abstract

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Dependence between coordinate extremes is a key factor in any multivariate risk assessment. Hence, it is of interest to know whether the components of a given multivariate random vector exhibit asymptotic independence or asymptotic dependence. In the latter case the structure of the asymptotic dependence has to be clarified. In the multivariate setting it is common to have an explicit form of the density rather than the distribution function. In this paper we therefore give criteria for asymptotic dependence in terms of the density. We consider distributions with light tails and restrict attention to continuous unimodal densities defined on the whole space or on an open convex cone. For simplicity, the density is assumed to be homothetic: all level sets have the same shape. Balkema and Nolde (2010) contains conditions on the shape which guarantee asymptotic independence. The situation for asymptotic dependence, treated in the present paper, is more delicate.

Keywords

Asymptotic dependencehomothetic level setmultivariate extremelight-tailed densitygeometric approach

MSC classification

Primary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 506 - 527
Copyright
© Applied Probability Trust 

References

Balkema, A. A. and Embrechts, P. (2007). High Risk Scenarios and Extremes. European Mathematical Society, Zürich.Google Scholar
Balkema, A. A. and Nolde, N. (2010). Asymptotic independence for unimodal densities. Adv. Appl. Prob. 42, 411–432.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Sibuya, M. (1960). Bivariate extreme statistics, I. Ann. Inst. Statist. Math. 11, 195–210.Google Scholar