  Correlated random fields are a common way to model dependence struc- tures in
high-dimensional data, especially for data collected in imaging. One important
parameter characterizing the degree of dependence is the asymp- totic variance
which adds up all autocovariances in the temporal and spatial domain.
Especially, it arises in the standardization of test statistics based on
partial sums of random fields and thus the construction of tests requires its
estimation. In this paper we propose consistent estimators for this parameter
for strictly stationary {\phi}-mixing random fields with arbitrary dimension of
the domain and taking values in a Euclidean space of arbitrary dimension, thus
allowing for multivariate random fields. We establish consistency, provide cen-
tral limit theorems and show that distributional approximations of related test
statistics based on sample autocovariances of random fields can be obtained by
the subsampling approach. As in applications the spatial-temporal correlations
are often quite local, such that a large number of autocovariances vanish or
are negligible, we also investigate a thresholding approach where sample
autocovariances of small magnitude are omitted. Extensive simulation studies
show that the proposed estimators work well in practice and, when used to
standardize image test statistics, can provide highly accurate image testing
procedures.
