  We consider the clustering of extremes for stationary regularly varying
random fields over arbitrary growing index sets. We study sufficient
assumptions on the index set such that the limit of the point random fields of
the exceedances above a high threshold exists. Under the so-called
anti-clustering condition, the extremal dependence is only local. Thus the
index set can have a general form compared to previous literature [3, 21].
However, we cannot describe the clustering of extreme values in terms of the
usual spectral tail measure [23] except for hyperrectangles or index sets in
the lattice case. Using the recent extension of the spectral measure for
star-shaped equipped space [18], the $\upsilon$-spectral tail measure provides
a natural extension that describes the clustering effect in full generality.
