2012/05/07 by Strokorb, Kirstin, Schlather, Martin
#60E15 (Secondary) #60G70 (Primary) 60G60 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1205.1315
We focus on two dependency quantities of a max-stable random field X on some space T: the extremal coefficient function θ which we define on finite sets of T and the extremal correlation function χ(s,t)=limx \uparrow ∞ \PP(Xs ≥ x | Xt ≥ x). We fully characterize extremal coefficient functions θ by a property called complete alternation and construct a corresponding max-stable random field. Simple properties and consequences concerning the convex geometry of extremal coefficients are derived. We study how the continuity of X, θ and χ are linked to each other, and we show that extremal correlation functions χ allow for convex combinations in general, and for products and pointwise limits if the resulting function is continuous. These are operations which are well-known for positive definite functions, but the latter are non-trivial for extremal correlation functions. Finally, we regard some additional implications, when the random field X on T=ℝd is stationary.