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Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution

The Extremal Index is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic processes with an Extremal Index for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised at at least two points of the phase space, where one of them is an indifferent periodic point and another one is either a repelling periodic point or a non periodic point. The occurrence of extreme events is then tied to the entrance and recurrence to the vicinities of those points. This enables to mix the behaviour of an Extremal Index equal to $0$ with that of an Extremal Index larger than $0$. Using bi-dimensional point processes we explain how mass escapes in order to destroy the usual relation. We also perform a study about the formulae to compute the limiting cluster size distribution introduced in \cite{FFT13,AFV15} and prove that ergodicity is enough to establish that the reciprocal of the Extremal Index is equal to the limit of the mean of the finite time cluster size distribution, which, in the case of the counterexamples given, does not coincide with the mean of the limit of the cluster size distribution.