Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T13:54:58.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakening the independence assumption on polar components: limit theorems for generalized elliptical distributions

Published online by Cambridge University Press:  24 March 2016

Miriam Isabel Seifert
Get access Rights & Permissions [Opens in a new window]

Abstract

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

Keywords

Conditional extreme value modelpolar representationelliptical distributionGumbel max-domain of attractionrandom norming
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 130 - 145
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdous, B., Fougères, A.-L. and Ghoudi, K. (2005). Extreme behaviour for bivariate elliptical distributions. Canad. J. Statist. 33, 317–334. CrossRefGoogle Scholar
Balkema, G. and Embrechts, P. (2007). High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society, Zürich. CrossRefGoogle Scholar
Berman, S. M. (1983). Sojourns and extremes of Fourier sums and series with random coefficients. Stoch. Process. Appl. 15, 213–238. CrossRefGoogle Scholar
Das, B. and Resnick, S. I. (2011). Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17, 226–252. CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. CrossRefGoogle Scholar
Fougères, A.-L. and Soulier, P. (2010). Limit conditional distributions for bivariate vectors with polar representation. Stoch. Models 26, 54–77. CrossRefGoogle Scholar
Geluk, J. L. and de Haan, L. (1987). Regular Variation, Extensions and Tauberian Theorems (CWI Tract 40). Stichting Mathematisch Centrum, Amsterdam. Google Scholar
Hashorva, E. (2012). Exact tail asymptotics in bivariate scale mixture models. Extremes 15, 109–128. CrossRefGoogle Scholar
Heffernan, J. E. and Resnick, S. I. (2007). Limit laws for random vectors with an extreme component. Ann. Appl. Prob. 17, 537–571. CrossRefGoogle Scholar
Heffernan, J. E. and Tawn, J. A. (2004). A conditional approach for multivariate extreme values. J. R. Statist. Soc. B 66, 497–546. CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes (Appl. Prob. 4). Springer, New York. CrossRefGoogle Scholar
Seifert, M. I. (2014). On conditional extreme values of random vectors with polar representation. Extremes 17, 193–219. CrossRefGoogle Scholar