Extremal clustering under moderate long range dependence and moderately heavy tails

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