A blocks method is used to define clusters of extreme values in stationary time series. The cluster starts at the first large value in the block and ends at the last one. The block cluster measure (the point measure at clusters) encodes different aspects of extremal properties. Its limiting behaviour is handled by vague convergence, hence the set of test functions consists of bounded, shift-invariant functionals that vanish around zero. If unbounded or non shift-invariant functionals are considered, we may obtain convergence at a different rate, depending on the type of the functional and the block size (small vs. large blocks). There are two prominent examples of such functionals: the locations of large jumps and the cluster length. We obtain a comprehensive characterization of the limiting behaviour of the block cluster measure evaluated at such functionals for stationary, regularly varying time series. Once the convergence of the block cluster measure is established, we can proceed with consistency of the empirical cluster measure. Consistency holds in the small and moderate blocks scenario, while fails in the large blocks situation. Next, we continue with weak convergence of the empirical cluster processes. The starting point is the seminal paper by Drees and Rootzen (2010). Under the appropriate uniform integrability condition (related to small blocks) the results in the latter paper are still valid. In the moderate and large blocks scenario, the Drees and Rootzen empirical cluster process diverges, but converges weakly when re-normalized properly.

翻译：采用分块方法定义平稳时间序列中极值簇。簇始于分块内首个大值，终于分块内末个大值。分块簇测度（簇的点测度）编码了极值性质的不同方面。其极限行为通过弱收敛处理，因此测试函数集由有界、平移不变且在零点附近消失的泛函构成。若考虑无界或非平移不变泛函，可能需以不同速率收敛，具体取决于泛函类型与分块规模（小分块与大分块）。此类泛函的两个典型例子为大跳跃位置与簇长度。我们针对平稳正则变化时间序列，全面刻画了分块簇测度在此类泛函作用下的极限行为。建立分块簇测度收敛性后，可进一步研究经验簇测度的相合性：该相合性在小分块与中等分块场景下成立，但在大分块场景下失效。继而研究经验簇过程的弱收敛，其起点为Drees与Rootzen（2010）的奠基性论文。在适当的均匀可积性条件（与小分块相关）下，该论文的结论仍然成立。在中等分块与大分块场景中，Drees-Rootzen经验簇过程发散，但经恰当重正则化后具有弱收敛性。