Exceedance probability of the integral of a stochastic process

Let X={X(s)}s∈SX={X(s)}s∈S be an almost sure continuous stochastic process (SS compact subset of RdRd) in the domain of attraction of some max-stable process, with index function constant over SS. We study the tail distribution of ∫SX(s)ds∫SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫SX(s)ds>x)P(∫SX(s)ds>x) for some high value xx, based on independent and identically distributed copies of XX. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. XX represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.