Shift-invariant homogeneous classes of random fields

11/01/2021
โˆ™
by   Enkelejd Hashorva, et al.
โˆ™
0
โˆ™

Let |ยท|:โ„^d โ†’ [0,โˆž) be a 1-homogeneous continuous map and let ๐’ฏ=โ„^l or ๐’ฏ=โ„ค^l with d,l positive integers. For a given โ„^d-valued random field (rf) Z(t),tโˆˆ๐’ฏ, which satisfies ๐”ผ{ |Z(t)|^ฮฑ}โˆˆ [0,โˆž) for all tโˆˆ๐’ฏ and some ฮฑ>0 we define a class of rf's ๐’ฆ^+_ฮฑ[Z] related to Z via certain functional identities. In the case ๐’ฏ=โ„^l the elements of ๐’ฆ^+_ฮฑ[Z] are assumed to be quadrant stochastically continuous. If B^h Z โˆˆ๐’ฆ^+_ฮฑ[Z] for any hโˆˆ๐’ฏ with B^h Z(ยท)= Z(ยท -h), hโˆˆ๐’ฏ, we call ๐’ฆ^+_ฮฑ[Z] shift-invariant. This paper is concerned with the basic properties of shift-invariant ๐’ฆ^+_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•Œ, which is of particular interest for shift-representations. Two applications of our findings concern max-stable rf's and their extremal indices.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset