Sparse regular variation

Abstract Regular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, spectral measure characterizes dependence structure of extremes. This gathers information on localization extreme events and often has sparse support since severe do not simultaneously occur in all directions. However, it is defined through weak convergence, which does provide natural way to capture this sparsity structure. paper, we introduce notion regular variation, makes possible better learn concept based Euclidean projection onto simplex, efficient algorithms are known. We prove that under mild assumptions equivalent notions, establish several results sparsely regularly varying random vectors.