Central limit theory for Peaks-over-Threshold partial sums of long memory linear time series

TL;DR

This paper develops new central limit theorems for Peaks-over-Threshold partial sums in long memory linear time series, especially when thresholds depend on sample size and innovations may have infinite variance.

Contribution

It extends asymptotic techniques to handle sample size-dependent thresholds and infinite variance innovations, providing new theoretical results for extreme value analysis.

Findings

Asymptotic normality holds for Peaks-over-Threshold estimators under new conditions.

Results apply to both heavy-tailed and light-tailed regimes.

Simulation confirms finite-sample relevance of theoretical findings.

Abstract

Over the last 30 years, extensive work has been devoted to developing central limit theory for partial sums of subordinated long memory linear time series. A much less studied problem, motivated by questions that are ubiquitous in extreme value theory, is the asymptotic behavior of such partial sums when the subordination mechanism has a threshold depending on sample size, so as to focus on the right tail of the time series. This article substantially extends longstanding asymptotic techniques by allowing the subordination mechanism to depend on the sample size in this way and to grow at a polynomial rate, while permitting the innovation process to have infinite variance. The cornerstone of our theoretical approach is a tailored reduction principle, which enables the use of classical results on partial sums of long memory linear processes. In this way we obtain asymptotic theory for certain Peaks-over-Threshold estimators with deterministic or random thresholds. Applications cover both heavy- and light-tailed regimes, yielding unexpected results which, to the best of our knowledge, are new to the literature. A simulation study illustrates the relevance of our findings in finite samples.