By Gaspard Monge

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Let Z1 ; Z2 ; : : : be independent observations from the distribution of Z. Choose yn so that P fY yn g 1=n. Z H / ) W as above. J / < 1: (8) n D n˛n . / ! The restriction of to HC is a probability measure, the distribution of W . 6. It is a central result. The first limit relation is analytical. It raises questions such as: 1) What limit laws are possible? 2) For a given limit law, what conditions on the distribution of Z will yield convergence? Preview 19 The second relation is more geometric.

1; (22) where Z r is the vector Z conditioned to lie outside the open ball rB. In this situation it is natural to use polar coordinates and write Z D R with R D kZk. The distribution of . ; R=r/, conditional on R r, converges to a product measure d dG on 26 Preview @B Œ1; 1/, where is the spectral measure, and G a Pareto distribution on Œ1; 1/ with density =r C1 , D 1= . The spectral measure gives an idea of the directions in which the data extremes cluster; the parameter in (21) describes the decay rate of the tails.

Our theory is based on concepts like scale invariance, self-similarity and symmetry. It is geometric and local. Independence is a global analytic assumption. It allows one to draw farreaching conclusions about extremes, but the techniques are different from those developed here. B/. These situations yield a limit measure with a one-parameter family of symmetries, the excess measure described in (9). Let us now return to high risk scenarios Z H where the halfspaces H are allowed to diverge in any direction.