Since many environmental processes such as heat waves or precipitation are spatial in extent, it is likely that a single extreme event affects several locations and the areal modelling of extremes is therefore essential if the spatial dependence of extremes has to be appropriately taken into account. This paper proposes a framework for conditional simulations of max-stable processes and give closed forms for Brown-Resnick and Schlather processes. We test the method on simulated data and give an application to extreme rainfall around Zurich and extreme temperature in Switzerland. Results show that the proposed framework provides accurate conditional simulations and can handle real-sized problems.

This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a maxstable (or max-infinitely divisible) process {η(t)} t∈T given observations {η(t i ) = y i , 1 ≤ i ≤ k}. Our starting point is the point process representation of maxinfinitely divisible processes by Giné, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. This paper offers new tools and perspective for prediction in extreme value theory together with numerous potential applications.

Let η = (η(t)) t∈T be a sample continuous max-infinitely random field on a locally compact metric space T . For a closed subset S ∈ T , we note η S the restriction of η to S. We consider β(S 1 , S 2 ) the absolute regularity coefficient between η S 1 and η S 2 , where S 1 , S 2 are two disjoint closed subsets of T . Our main result is a simple upper bound for β(S 1 , S 2 ) involving the exponent measure µ of η: we prove thatIf η is a simple max-stable random field, the upper bound is related to the so-called extremal coefficients: for countable disjoint sets S 1 and S 2 , we obtain β(S 1 , S 2 ) ≤ 4 (s 1 ,s 2 )∈S 1 ×S 2 (2 − θ(s 1 , s 2 )), where θ(s 1 , s 2 ) is the pair extremal coefficient.As an application, we show that these new estimates entail a central limit theorem for stationary max-infinitely divisible random fields on Z d . In the stationary maxstable case, we derive the asymptotic normality of three simple estimators of the pair extremal coefficient.Key words: absolute regularity coefficient; max-infinitely divisible random field; maxstable random field; central limit theorem for weakly dependent random field.