  The asymptotic results that underlie applications of extreme random fields
often assume that the variables are located on a regular discrete grid,
identified with $\mathbb{Z}^2$, and that they satisfy stationarity and isotropy
conditions. Here we extend the existing theory, concerning the asymptotic
behavior of the maximum and the extremal index, to non-stationary and
anisotropic random fields, defined over discrete subsets of $\mathbb{R}^2$. We
show that, under a suitable coordinatewise long range dependence condition, the
maximum may be regarded as the maximum of an approximately independent sequence
of submaxima, although there may be high local dependence leading to clustering
of high values. Under restrictions on the local path behavior of high values,
criteria are given for the existence and value of the spatial extremal index
which plays a key role in determining the cluster sizes and quantifying the
strength of dependence between exceedances of high levelsThe general theory is
applied to the class of max-stable random fields, for which the extremal index
is obtained as a function of well-known tail dependence measures found in the
literature, leading to a simple estimation method for this parameter. The
results are illustrated with non-stationary Gaussian and 1-dependent random
fields. For the latter, a simulation and estimation study is performed.
