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Extremal clusters of stationary processes with long memory can be quiteintricate. For certain stationary infinitely divisible processes withsubexponential tails, including both power-like tails and certain lightertails, e.g. lognormal-like tails, such clusters may take the shape of stableregenerative sets. In this paper we show that for semi-exponential tails, whichare even lighter, a new shape of extremal clusters arises. In this case eachstable regenerative set supports a random panoply of varying extremes.

A New Shape of Extremal Clusters For Certain Stationary Semi-Exponential Processes With Moderate Long Range Dependence