Limit theorems for unbounded cluster functionals of regularly varying time series

TL;DR

This paper investigates the limiting behavior of cluster functionals in stationary, regularly varying time series, especially unbounded and non shift-invariant functionals, and discusses the implications for empirical cluster measures.

Contribution

It provides a comprehensive characterization of the limits of unbounded cluster functionals and analyzes the convergence properties of empirical cluster measures under various block size scenarios.

Findings

Limiting behavior characterized for unbounded functionals like jump locations and cluster length.

Empirical cluster measure is consistent in small and moderate blocks, but not in large blocks.

Weak convergence of empirical cluster processes requires re-normalization in moderate and large blocks.

Abstract

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.