Clustering of large deviations events in heavy-tailed moving average processes: the catastrophe principle in the short-memory case

TL;DR

This paper investigates how large deviation events cluster in short-memory heavy-tailed moving average processes, revealing a distinct 'catastrophe' principle that differs from the light-tailed case.

Contribution

It demonstrates that the clustering structure of large deviations in heavy-tailed short-memory processes follows a different principle than in light-tailed cases, highlighting the role of tail heaviness.

Findings

Large deviations cluster differently in heavy-tailed processes.

The 'catastrophe' principle governs large deviations in heavy tails.

Contrast with light-tailed processes shows different clustering mechanisms.

Abstract

How do large deviation events in a stationary process cluster? The answer depends not only on the type of large deviations, but also on the length of memory in the process. Somewhat unexpectedly, it may also depend on the tails of the process. In this paper we work in the context of large deviations for partial sums in moving average processes with short memory and regularly varying tails. We show that the structure of the large deviation cluster in this case markedly differs from the corresponding structure in the case of exponentially light tails, considered in Chakrabarty and Samorodnitsky (2024). This is due to the difference between the ``conspiracy'' vs. the ``catastrophe'' principles underlying the large deviation events in the light tailed case and the heavy tailed case, correspondingly.