Principal component analysis for multivariate extremes

In the probabilistic framework of multivariate regular variation, first order behavior heavy-tailed random vectors above large radial thresholds is ruled by a homogeneous limit measure. For high dimensional vector, reasonable assumption that support this measure concentrated on lower subspace, meaning certain linear combinations components are much likelier to be than others. Identifying subspace and thus reducing dimension will facilitate refined statistical analysis. work we apply Principal Component Analysis (PCA) re-scaled version radially thresholded observations. Within learning empirical risk minimization, our main focus analyze squared reconstruction error for exceedances over thresholds. We prove converges true risk, uniformly all projection subspaces. As consequence, best shown converge in probability optimal one, terms Hausdorff distance between their intersections with unit sphere. addition, if ball, obtain finite sample uniform guarantees pertaining estimated subspace. Numerical experiments illustrate capability proposed improve estimators extreme value parameters.