Clustering of large deviations in heavy-tailed moving averages: the catastrophe principle in the long-memory case

Abstract

Clustering of large deviations events in a stationary stochastic process depends critically on the interplay between the tail behavior of the marginal distribution and the strength of temporal dependence. In the class of doubly infinite moving average processes, when the memory is short, previous work has established a sharp contrast in the clustering patterns between light- and heavy-tailed settings, governed by \textit{conspiracy} and \textit{catastrophe} principles respectively. It has also been described how long memory interacts with the conspiracy principle to affect clustering in the light-tailed case. This paper addresses the interaction of long memory with the catastrophe principle in the heavy-tailed case. It turns out that long memory generally allows for a wider range of catastrophes to play a role and leads to longer and qualitatively different clustering patterns.