Limit Theory for the Sample Autocovariance for Heavy Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows

TL;DR

This paper establishes limit theorems for sample autocovariances of heavy-tailed, stationary infinitely divisible processes generated by conservative flows, revealing how tail heaviness and memory influence their growth rates.

Contribution

It introduces a general framework using infinite ergodic theory to analyze the asymptotics of autocovariances in heavy-tailed processes generated by conservative flows.

Findings

Growth rate depends on tail heaviness and process memory

Provides a unified approach for a broad class of processes

Extends previous specific process results

Abstract

This study aims to develop the limit theorems on the sample autocovariances and sample autocorrelations for certain stationary infinitely divisible processes. We consider the case where the infinitely divisible process has heavy tail marginals and is generated by a conservative flow. Interestingly, the growth rate of the sample autocovariances is determined by not only heavy tailedness of the marginals but also memory length of the process. Although this feature was first observed by \cite{resnick:samorodnitsky:xue:2000} for some very specific processes, we will propose a more general framework from the viewpoint of infinite ergodic theory. Consequently, the asymptotics of the sample autocovariances can be more comprehensively discussed.