Shift-invariant homogeneous classes of random fields

11/01/2021
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by   Enkelejd Hashorva, et al.
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Let |Β·|:ℝ^d β†’ [0,∞) be a 1-homogeneous continuous map and let 𝒯=ℝ^l or 𝒯=β„€^l with d,l positive integers. For a given ℝ^d-valued random field (rf) Z(t),tβˆˆπ’―, which satisfies 𝔼{ |Z(t)|^Ξ±}∈ [0,∞) for all tβˆˆπ’― and some Ξ±>0 we define a class of rf's 𝒦^+_Ξ±[Z] related to Z via certain functional identities. In the case 𝒯=ℝ^l the elements of 𝒦^+_Ξ±[Z] are assumed to be quadrant stochastically continuous. If B^h Z βˆˆπ’¦^+_Ξ±[Z] for any hβˆˆπ’― with B^h Z(Β·)= Z(Β· -h), hβˆˆπ’―, we call 𝒦^+_Ξ±[Z] shift-invariant. This paper is concerned with the basic properties of shift-invariant 𝒦^+_Ξ±[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 π•Œ, 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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