We study clusters of rare events, namely, exceedances of the sequence over a sufficiently high
threshold. Such clusters are caused by dependence in time series. Such clusters of rare events
and the asymptotic distributions of the cluster and inter-cluster sizes have been widely studied
due to numerous applications, see Ancona-Navarrete and Tawn (2000), Beirlant et al. (2004),
Ferro and Segers (2003), Markovich (2014), Robert (2009), (2013), Robinson et al. (2000)
among others. We define the clusters as conglomerates containing consecutive threshold ex-
ceedances of the process separated by return intervals with consecutive non-exceedances. We
establish geometric-like asymptotical distributions of the cluster and inter-cluster sizes and
corresponding expectations. Rates of convergence of the distribution tail of the duration of
clusters and return intervals between clusters to stable distribution tail are derived. The talk
is based on recent author results. Distributions of the cluster and inter-cluster sizes of some
example processes are presented. Applications of the results in social and telecommunication
systems are discussed.
