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.

Published J. Mult. Analysis Applications

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• Stationary max-stable fields associated to negative definite functions
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• The tail process and tail measure of continuous time regularly varying stochastic processes
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Cited by in corpus (1)

• Cluster Random Fields and Random-Shift Representations