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On the foundations of multivariate heavy-tail analysis

Part of: Limit theorems Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Sidney Resnick
Sidney Resnick*
Affiliation:
School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: sid@orie.cornell.edu
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Abstract

Univariate heavy-tailed analysis rests on the analytic notion of regularly varying functions. For multivariate heavy-tailed analysis, reliance on functions is awkward because multivariate distribution functions are not natural objects for many purposes and are difficult to manipulate. An approach based on vague convergence of measures makes the differences between univariate and multivariate analysis evaporate. We survey the foundations of the subject and discuss statistical attempts to assess dependence of large values. An exploratory technique is applied to exchange rate return data and shows clear differences in the dependence structure of large values for the Japanese Yen versus German Mark compared with the French Franc versus the German Mark.

Keywords

Regular variationextremal dependenceangular measureheavy tailsasymptotic independence

MSC classification

Primary: 62G32: Statistics of extreme values; tail inference
Secondary: 60F05: Central limit and other weak theorems
Type
Part 4. Heavy-tail analysis
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 191 - 212
Copyright
Copyright © Applied Probability Trust 2004 

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