Shift-invariant homogeneous classes of random fields

11/01/2021

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by   Enkelejd Hashorva, et al.

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Let |·|:ℝ^d → [0,∞) be a 1-homogeneous continuous map and let 𝒯=ℝ^l or 𝒯=ℤ^l with d,l positive integers. For a given ℝ^d-valued random field (rf) Z(t),t∈𝒯, which satisfies 𝔼{ |Z(t)|^α}∈ [0,∞) for all t∈𝒯 and some α>0 we define a class of rf's 𝒦^+_α[Z] related to Z via certain functional identities. In the case 𝒯=ℝ^l the elements of 𝒦^+_α[Z] are assumed to be quadrant stochastically continuous. If B^h Z ∈𝒦^+_α[Z] for any h∈𝒯 with B^h Z(·)= Z(· -h), h∈𝒯, we call 𝒦^+_α[Z] shift-invariant. This paper is concerned with the basic properties of shift-invariant 𝒦^+_α[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 𝕌, which is of particular interest for shift-representations. Two applications of our findings concern max-stable rf's and their extremal indices.