Asymptotics for irregularly observed long memory processes

TL;DR

This paper investigates how irregular observation times, modeled by a renewal process, influence the asymptotic behavior of the sample mean of long memory processes, revealing a phase transition based on the tail properties of the renewal process.

Contribution

It characterizes the asymptotic distribution of the sample mean under irregular sampling, showing a transition from normal to mixture distributions depending on the renewal process tail behavior.

Findings

For heavy-tailed renewal times, the limit is a Normal Variance Mixture.

For lighter tails, the limit is asymptotically normal.

The limiting variance in the heavy-tailed case is characterized by a Lévy stable motion.

Abstract

We study the effect of observing a stationary process at irregular time points via a renewal process. We establish a sharp difference in the asymptotic behaviour of the self-normalized sample mean of the observed process depending on the renewal process. In particular, we show that if the renewal process has a moderate heavy tail distribution then the limit is a so-called Normal Variance Mixture (NVM) and we characterize the randomized variance part of the limiting NVM as an integral function of a L\'evy stable motion. Otherwise, the normalized sample mean will be asymptotically normal.