Convergence of clusters

Abstract

When a regularly varying stationary time series is extremally dependent, then the exceedences over high levels will tend to happen in clusters. From a statistical point of view, it is important to identify these clusters in order to make inference on the extremal dependence of the time series. A very common statistical practice is, given n successive observations of the said time series, to split them into $m_n$ blocks of size $r_n$ (with $m_nr_n\approx n$ and $m_n, r_n\rightarrow \infty$), compute a given extreme value statistics on each of the $m_n$ blocks, and average these values to obtain an estimator. The simplest example is the number of exceedences over a high threshold $c_n$: in each block the number of exceedences will be recorded, and the average of these numbers will be an estimator of the mean cluster size. We expect that the statistics computed on each block converge to the corresponding statistics computed on the tail process, which can be seen as an asymptotic representation for the block values scaled by $c_n$ when the block size $r_n$ and the threshold $c_n$ both tend to infinity in an appropriate way.

6.4 Bibliographical notes

This chapter is essentially based on [BPS18] and [PS18]. The main ideas on convergence of cluster and cluster indices (or functionals) stem from [BS09] where earlier references on cluster functionals can be found, most notably [Yun00] and [Seg03].

Corollary 6.2.6 is taken from Proposition 4.2 of [BS09], whose proof is the model for the proof of Theorem 6.2.5. Section 6.2 extends results of [MW14, MW16].