  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.
