  Peaks-over-threshold analysis using the generalized Pareto distribution is
widely applied in modelling tails of univariate random variables, but much
information may be lost when complex extreme events are studied using
univariate results. In this paper, we extend peaks-over-threshold analysis to
extremes of functional data. Threshold exceedances defined using a functional
$r$ are modelled by the generalized $r$-Pareto process, a functional
generalization of the generalized Pareto distribution that covers the three
classical regimes for the decay of tail probabilities, and that is the only
possible continuous limit for $r$-exceedances of a properly rescaled process.
We give construction rules, simulation algorithms and inference procedures for
generalized $r$-Pareto processes, discuss model validation, and use the new
methodology to study extreme European windstorms and heavy spatial rainfall.
