  Central limit theorems are established for the sum, over a spatial region, of
observations from a linear process on a $d$-dimensional lattice. This region
need not be rectangular, but can be irregularly-shaped. Separate results are
established for the cases of positive strong dependence, short range
dependence, and negative dependence. We provide approximations to asymptotic
variances that reveal differential rates of convergence under the three types
of dependence. Further, in contrast to the one dimensional (i.e., the time
series) case, it is shown that the form of the asymptotic variance in
dimensions $d>1$ critically depends on the geometry of the sampling region
under positive strong dependence and under negative dependence and that there
can be non-trivial edge-effects under negative dependence for $d>1$. Precise
conditions for the presence of edge effects are also given.
