Extremal clustering in non-stationary random sequences
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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. The extent to which extremal clustering may occur is measured by the extremal index. Here we consider non-stationary sequences subject to suitable long range dependence restrictions. We first consider the case of non-independent but identically distributed random variables. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence and generalizes the result of O'Brien (1987) for the extremal index. Based on this new representation we derive the asymptotic distribution for the times between consecutive extreme observations and use this to construct moment and likelihood based estimators for parameters that measure the potential for extremal clustering at a particular location in the sequence. We also discuss extensions to the case of different marginal distributions.
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