Sojourn functionals for spatiotemporal random fields with long-memory

TL;DR

This paper investigates the asymptotic behavior of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence, deriving reduction theorems and analyzing convergence to Rosenblatt-type distributions.

Contribution

It provides new reduction theorems for nonlinear functionals of long-memory spatiotemporal Gaussian fields under general covariance structures.

Findings

Reduction theorems for local functionals of nonlinear transformations.

Convergence of Minkowski functionals to Rosenblatt-type distributions.

Applicability to Gneiting class covariance structures.

Abstract

This paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank m larger than or equal to 1, under general covariance structures. These results are proven to hold, in particular, for a family of non--separable covariance structures belonging to Gneiting class. For m=2, under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt type limiting distribution for a suitable range of the long memory parameter.