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On the foundations of multivariate heavy-tail analysis. (English) Zbl 1049.62056

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 purpuses and are difficult to manipulate.

An approach based on vague convergence of measures makes the differences between univariute 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.

MSC:
62G32Statistics of extreme values; tail inference
62P05Applications of statistics to actuarial sciences and financial mathematics
62H99Multivariate analysis
60F05Central limit and other weak theorems
62H05Characterization and structure theory (Multivariate analysis)