Shift-invariant homogeneous classes of random fields

Abstract

Given an $R^d$-valued random field (rf) $Z(t),t\in T$ and an $\alpha$-homogeneous mapping $\kappa$ we define the corresponding equivalent class of rf's (denoted by $K_\alpha$) which include representers of the same tail measure $\nu_Z$. When $T$ is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of $Z$. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant $K_\alpha$'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps $U$ acting on elements of $K_\alpha$ turns out to be crucial for properties of functionals of $Z$. Applications of our findings concern max-stable and symmetric $\alpha$-stable rf's, their maximal indices as well as their random shift-representations.