Characterizing extremal coefficient functions and extremal correlation functions

Abstract

We focus on two dependency quantities of a max-stable random field $X$ on some space $T$: the extremal coefficient function $\theta$ which we define on finite sets of $T$ and the extremal correlation function $\chi(s,t)=\lim_{x \uparrow \infty} \PP(X_s \geq x \mid X_t \geq x)$. We fully characterize extremal coefficient functions $\theta$ 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 $X$, $\theta$ and $\chi$ are linked to each other, and we show that extremal correlation functions $\chi$ 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 $T=\mathbb{R}^d$ is stationary.