Clustering indices and decay of correlations in non-Markovian models

TL;DR

This paper investigates clustering phenomena and correlation decay in non-Markovian models, emphasizing the importance of finite-time analysis over asymptotic limits for understanding cluster sizes and related indicators.

Contribution

It introduces a general regenerative process framework that includes Smith's model, highlighting the significance of finite-time quantities and comparing different cluster size indicators.

Findings

Finite-time quantities provide better insights into cluster sizes.

The study shows differences between asymptotic and finite-time behaviors.

Decay of correlations in non-Markovian models is characterized.

Abstract

When there is no independence, abnormal observations may have a tendency to appear in clusters instead of scattered along the time frame. Identifying clusters and estimating their size are important problems arising in statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the Extremal Index appears associated to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing the limiting point processes. Smith \cite{S88} introduced a regenerative process exhibiting such disagreement. Very recently, in \cite{AFF18} the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the Extremal Index. Here, we consider a general regenerative process that includes Smith's model and show that it is important to consider finite time quantities instead of asymptotic ones and compare their different behaviours in relation to the cluster size. We consider other indicators such as what we call the sojourn time, which corresponds to the size of groups of abnormal observations, when there is some uncertainty regarding where the cluster containing that group was actually initiated. We also study the decay of correlations of the non-Markovian models considered.