  It is well known that the distribution of extreme values of strictly
stationary sequences differ from those of independent and identically
distributed sequences in that extremal clustering may occur. Here we consider
non-stationary but identically distributed sequences of random variables
subject to suitable long-range dependence restrictions. We find that the
limiting distribution of appropriately normalized sample maxima depends on a
parameter that measures the average extremal clustering of the sequence. Based
on this new representation we derive the asymptotic distribution for the time
between consecutive extreme observations and construct moment and
likelihood-based estimators for measures of extremal clustering. We specialize
our results to random sequences with periodic dependence structure.
