Asymptotics for irregularly observed long memory processes

Abstract

We study the effect of observing a long-memory 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 Lévy stable motion. Otherwise, the normalized sample mean will be asymptotically normal.