  We study clustering of the extremes in a stationary sequence with
subexponential tails in the maximum domain of attraction of the Gumbel We
obtain functional limit theorems in the space of random sup-measures and in the
space $D(0,\infty)$. The limits have the Gumbel distribution if the memory is
only moderately long. However, as our results demonstrate rather strikingly,
the "heuristic of a single big jump" could fail even in a moderately long range
dependence setting. As the tails become lighter, the extremal behavior of a
stationary process may depend on multiple large values of the driving noise.
