Shift-invariant homogeneous classes of random fields

Let $|\cdot|:\mathbb{R}^d \to [0,\infty) $ be a $1$-homogeneous continuous map and let $\mathcal{T}=\mathbb{R}^l$ or $\mathcal{T}=\mathbb{Z}^l$ with $d,l$ positive integers. For a given $\mathbb{R}^d$-valued random field (rf) $Z(t),t\in \mathcal{T}$, which satisfies $\mathbb{E}\{ |Z(t)|^\alpha\} \in [0,\infty)$ for all $t\in \mathcal{T}$ and some $\alpha>0$ we define a class of rf's $\mathcal{K}^+_\alpha[Z]$ related to $Z$ via certain functional identities. In the case $\mathcal{T}=\mathbb{R}^l$ the elements of $\mathcal{K}^+_\alpha[Z]$ are assumed to be quadrant stochastically continuous. If $B^h Z \in \mathcal{K}^+_\alpha[Z]$ for any $h\in \mathcal{T}$ with $B^h Z(\cdot)= Z(\cdot -h), h\in \mathcal{T}$, we call $\mathcal{K}^+_\alpha[Z]$ shift-invariant. This paper is concerned with the basic properties of shift-invariant $\mathcal{K}^+_\alpha[Z]$'s. In particular, we discuss functional equations that characterise the shift-invariance and relate it with spectral tail and tail rf's introduced in this article for our general settings. Further, we investigate the class of universal maps $\mathbb{U}$, which is of particular interest for shift-representations. Two applications of our findings concern max-stable rf's and their extremal indices.