High-level moving excursions for spatiotemporal Gaussian random fields with long range dependence

This paper analyzes the limit distribution of spatiotemporal sojourn measures of Gaussian subordinated random fields. The family of Gaussian spatiotemporal random fields studied displays long--range dependence. The approach presented is based on Hermite expansion, and reduction theorems. Two new ingredients are incorporated to this analysis. Specifically, on the one hand, the limiting distribution is computed under increasing domain asymptotics in space and/or time, and, on the other hand, the involved non--linear transformation in the Gaussian subordination is time dependent. In particular, a central limit theorem is obtained for first Minkowski functional providing the spatiotemporal volume of excursion sets of spatiotemporal Gaussian random fields under moving levels. These results are also applied to the case where subordination to a spherical Gaussian random field, defined by restriction to the sphere, is performed.