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On limiting cluster size distributions for processes of exceedances for stationary sequences

It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate that, for any given distribution G on the natural numbers, there exists a stationary sequence for which the compounding law of this limiting process of exceedances will coincide with G.