Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs. also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example. efulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio ∑ i ⩽ d a i U i with positive weights a i , i ≤ d , is independent of the weights, if ( U 1 , … , U d ) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261–290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.