  Extremes occur in stationary regularly varying time series as short periods
with several large observations, known as extremal blocks. We study cluster
statistics summarizing the behavior of functions acting on these extremal
blocks. Examples of cluster statistics are the extremal index, cluster size
probabilities, and other cluster indices. The purpose of our work is twofold.
First, we state the asymptotic normality of block estimators for cluster
inference based on consecutive observations with large $\ell^p$-norms, for $p>
0$. The case $p=\alpha$, where $\alpha> 0$ is the tail index of the time
series, has specific nice properties thus we analyze the asymptotic of blocks
estimators when approximating $\alpha$ using the Hill estimator. Second, we
verify the conditions we require on classical models such as linear models and
solutions of stochastic recurrence equations. Regarding linear models, we prove
that the asymptotic variance of classical index cluster-based estimators is
null as first conjectured in Hsing T. [26]. We illustrate our findings on
simulations.
