  When there is no independence, abnormal observations may have a tendency to
appear in clusters instead of scattered along the time frame. Identifying
clusters and estimating their size are important problems arising in statistics
of extremes or in the study of quantitative recurrence for dynamical systems.
In the classical literature, the Extremal Index appears associated 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
the limiting point processes. Smith \cite{S88} introduced a regenerative
process exhibiting such disagreement. Very recently, in \cite{AFF18} 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 and 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.
