Clustering indices and decay of correlations in non-Markovian models

When there is no independence, abnormal observations may have a tendency to appear in clusters instead of being scattered along the time frame. Identifying clusters and estimating their size are important problems arising in the statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the extremal index appears to be linked to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing limiting point processes. Smith (1988 Adv. Appl. Probab. 20 681-3), introduced a regenerative process exhibiting such disagreement. Very recently, in Abadi (2018 (arXiv:1808.02970)) the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the extremal index. Here, we consider a general regenerative process that includes Smith's model, show that it is important to consider finite time quantities instead of asymptotic ones and compare their different behaviours in relation to the cluster size. We consider other indicators, such as what we call the sojourn time, which corresponds to the size of groups of abnormal observations, when there is some uncertainty regarding where the cluster containing that group was actually initiated. We also study the decay of correlations of the non-Markovian models considered.