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We consider disjoint and sliding blocks estimators of cluster indices for multivariate, regularly varying time series in the Peak-over-Threshold framework. We aim to provide a complete description of the limiting behaviour of these estimators. This is achieved by a precise expansion for the difference between the sliding and the disjoint blocks statistics. The rates in the expansion stem from internal clusters and boundary clusters. To obtain these rates we need to extend the existing results on vague convergence of cluster measures. We reveal dichotomous behaviour between small blocks and large blocks scenario., Comment: 52 pages

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., Comment: 35 pages

Extremal clusters of stationary processes with long memory can be quite intricate. For certain stationary infinitely divisible processes with subexponential tails, including both power-like tails and certain lighter tails, e.g. lognormal-like tails, such clusters may take the shape of stable regenerative sets. In this paper we show that for semi-exponential tails, which are even lighter, a new shape of extremal clusters arises. In this case each stable regenerative set supports a random panoply of varying extremes., Comment: 41 pages

We study clustering of the extremes in a stationary sequence with subexponential tails in the maximum domain of attraction of the Gumbel We obtain functional limit theorems in the space of random sup-measures and in the space $D(0,\infty)$. The limits have the Gumbel distribution if the memory is only moderately long. However, as our results demonstrate rather strikingly, the "heuristic of a single big jump" could fail even in a moderately long range dependence setting. As the tails become lighter, the extremal behavior of a stationary process may depend on multiple large values of the driving noise.

We study the extremes for a class of a symmetric stable random fields with long range dependence. We prove functional extremal theorems both in the space of sup measures and in the space of cadlag functions of several variables. The limits in both types of theorems are of a new kind, and only in a certain range of parameters these limits have the Fr\'{e}chet distribution.