Functional regular variations, Pareto processes and peaks over threshold

History: The latest developments of extreme value theory focus on the functional framework and much effort has been put in the theory of max-stable processes and functional regular variations. Paralleling the univariate extreme value theory, this work focuses on the exceedances of a stochastic process above a high threshold and their connections with generalized Pareto processes. More precisely we define an exceedance through a homogeneous cost functional l and show that the limiting (rescaled) distribution is a l-Pareto process whose spectral measure can be characterized. Three equivalent characterizations of the l-Pareto process are given using either a constructive approach, either a homogeneity property or a peak over threshold stability property. We also provide non parametric estimators of the spectral measure and give some examples.