Extremes in stochastic sequences
arising as clusters of exceedances over a threshold are observed in numerous applications of climate research, finance, telecommunication and social systems.
Such clusters are caused by dependence. We define the cluster as a conglomerate containing consecutive exceedances of the underlying process $\{R_n\}$ over a threshold $u$ separated by return intervals with consecutive non-exceedances. The geometric-like limit distributions of the cluster size %$T_2(x_{\rho_n})$
and inter-cluster size
are obtained in Markovich-2014 under the mixing condition similar to one used in \cite{Ferro}. The sequence of high $(1-\rho_n)$-order quantiles $x_{\rho_n}$ of the $\{R_n\}$ is used as thresholds. The obtained distributions differ from the geometric distribution by the extremal index of the process $\{R_n\}$. The asymptotic first moments of both cluster characteristics are obtained \cite{Markovich-2013}, \cite{Markovich-2014}.
Similarly, the result can be extended to moments of higher orders. In \cite{Markovich-2014} the duration of (inter-)clusters
is defined as a sum of the random number of weakly dependent, regularly varying inter-arrival times between events of interest with tail index $0
