On the asymptotics of extremal lp-blocks cluster inference

TL;DR

This paper investigates the asymptotic behavior of cluster statistics in stationary regularly varying time series, providing theoretical results and verifying conditions for classical models, with a focus on extremal blocks and their estimators.

Contribution

It establishes the asymptotic normality of block estimators for extremal cluster inference, especially for p<0 and p=α, and verifies conditions for classical models like linear processes.

Findings

Asymptotic normality of block estimators for extremal clusters.

Null asymptotic variance for classical index estimators in linear models.

Validation of theoretical conditions on models through simulations.

Abstract

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 lp-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.