Functional regular variations, Pareto processes and peaks over threshold.

Summary Although the last decades have seen many developments on max-stable processes, little is known on the limiting distribution of exceedances of stochastic processes. Paralleling the univariate extreme value theory, this work focuses on threshold exceedances of a stochastic process and their connections with regularly varying and generalized Pareto processes. More precisely we define an exceedance through a cost functional ` and show that the limiting (rescaled) distribution is a simple `–Pareto process whose spectral measure can be characterized. Several equivalent constructions for `–Pareto processes are given using either a constructive approach, either an homogeneity property or a peak over threshold stability. We also provide an estimator of the spectral measure and give some examples.