The extremal index (EI) 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 EI for which that equality does not hold. The mechanism used to build such counterexamples is based on considering observable functions maximised 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 EI equal to 0 with that of an EI 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 cluster size distribution introduced earlier and prove that ergodicity is enough to establish that the finite versions of the reciprocal of the EI and of the mean of the cluster size distribution do coincide.

中文翻译：

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关于极值指数和极限簇尺寸分布平均值的动态反例

极值指数（EI）是衡量稀有事件聚类强度的参数，通常等于极限聚类大小分布平均值的倒数。我们展示了如何使用不具有相等性的EI构建动态生成的随机过程。用于构建此类反例的机制是基于考虑使相空间的至少两个点最大化的可观察函数，其中一个是无关紧要的周期点，另一个是排斥周期点或非周期性点。然后，将极端事件的发生与这些点的进入和重复联系在一起。这样可以将EI等于0的行为与EI大于0的行为混合。使用二维点过程，我们解释了质量如何逃逸以破坏通常的关系。我们还对前面介绍的用于计算簇大小分布的公式进行了研究，并证明了遍历性足以证明EI倒数的有限形式和簇大小分布的均值的确是一致的。