We deal with clusters of exceedances of the underlying process $\{R_n\}$ over a threshold which are caused by dependence in time series. The asymptotically equal geometric distributions of cluster and inter-cluster sizes and the first moment of the cluster size are derived in Markovich (2013). The threshold is taken equal to a high quantile of $\{R_n\}$. The latter inferences are extended and asymptotic first two moments of both cluster characteristics are obtained.