Shift-invariant homogeneous classes of random fields
Let |ยท|:โ^d โ [0,โ) be a 1-homogeneous continuous map and let ๐ฏ=โ^l or ๐ฏ=โค^l with d,l positive integers. For a given โ^d-valued random field (rf) Z(t),tโ๐ฏ, which satisfies ๐ผ{ |Z(t)|^ฮฑ}โ [0,โ) for all tโ๐ฏ and some ฮฑ>0 we define a class of rf's ๐ฆ^+_ฮฑ[Z] related to Z via certain functional identities. In the case ๐ฏ=โ^l the elements of ๐ฆ^+_ฮฑ[Z] are assumed to be quadrant stochastically continuous. If B^h Z โ๐ฆ^+_ฮฑ[Z] for any hโ๐ฏ with B^h Z(ยท)= Z(ยท -h), hโ๐ฏ, we call ๐ฆ^+_ฮฑ[Z] shift-invariant. This paper is concerned with the basic properties of shift-invariant ๐ฆ^+_ฮฑ[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 ๐, which is of particular interest for shift-representations. Two applications of our findings concern max-stable rf's and their extremal indices.
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