  Motivated by examples from extreme value theory we introduce the general
notion of a cluster process as a limiting point process of returns of a certain
event in a time series. We explore general invariance properties of cluster
processes which are implied by stationarity of the underlying time series under
minimal assumptions. Of particular interest are the cluster size distributions,
where we introduce the two notions of inspected and typical cluster sizes and
derive general properties of and connections between them. While the extremal
index commonly used in extreme value theory is often interpreted as the inverse
of a "mean cluster size", we point out that this only holds true for the
expected value of the typical cluster size, caused by an effect very similar to
the inspection paradox in renewal theory.
