Shift-generated $\alpha$-homogeneous classes of jointly measurable random fields

TL;DR

This paper introduces a new class of shift-generated alpha-homogeneous random fields derived from a functional identity involving a measurable RF, which can generate max-stable and stationary RFs, and explores their properties and connections.

Contribution

It extends the functional identity defining shift-generated RFs to include broader functionals and proves the existence of L^alpha-continuous elements within this class.

Findings

C[Z] contains at least one L^alpha-continuous element.

The class generates max-stable and stationary RFs.

Properties of local RFs and their spectral tail connections are analyzed.

Abstract

We consider a class of shift-generated alpha-homogeneous random fields (RFs) C[Z] defined through a functional identity involving a fixed positive alpha and a given jointly measurable R^d-valued RF Z(t),t in R^l. The significance of such classes lies in the fact that their elements generate max-stable and stationary RFs. We extend the original functional identity to a broad class of functionals, including the integral operator S(.) and prove that C[Z] contains at least one L^alpha-continuous element. Finally, we investigate properties of local RFs and their connections with spectral tail and tail RFs.