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.

翻译：极端值出现在平稳正则变化时间序列中，表现为若干大观测值构成的短时段，即极值分块。我们研究总结作用于这些极值分块上函数行为的聚类统计量。聚类统计量的例子包括极值指数、聚类规模概率及其他聚类指数。本研究的目的有二。首先，对于基于具有大ℓ^p-范数的连续观测的分块估计量（p>0），我们陈述其在聚类推断中的渐近正态性。当p=α（α>0为时间序列的尾部指数）时，该情形具有特定优良性质，因此我们分析了在使用Hill估计量近似α时分块估计量的渐近性。其次，我们在经典模型（如线性模型和随机递归方程的解）上验证了所需条件。对于线性模型，我们证明经典聚类指数分块估计量的渐近方差为零——该结论最初由Hsing T. [26]提出猜想。我们通过数值模拟验证了研究结果。