In many applications it is important to evaluate the impact of clusters of observations caused by the dependence and heaviness of tails
in time series.
We consider a stationary sequence of random variables $\{R_n\}_{n\ge 1}$ with marginal cumulative distribution function $F(x)$ and the extremal index
$\theta\in[0,1]$.
The clusters contain consecutive exceedances of the time series over a threshold $u$ separated by return intervals with consecutive non-exceedances.
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We derive geometric forms of asymptotically equal distributions of the normalized cluster and inter-cluster sizes that depend on $\theta$. The inter-cluster size determines the number $T_1(u)$ of inter-arrival times between observations of the process $R_t$ arising between two consecutive clusters. The cluster size is equal to the number $T_2(u)$ of inter-arrival times within clusters. The inferences are valid when $u$ is taken as a sufficiently high quantile of the process $\{R_n\}$.
The derived geometric models allow us to obtain the asymptotically equal means of $T_1(u)$ and $T_2(u)$ and other indices of clusters.
