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