Asymptotics for irregularly observed long memory processes

2024/09/14 by Mohamedou Ould-Haye, Ould-Haye, Mohamedou, Anne Philippe +1
Computer Science · Engineering · #FOS: Mathematics #Fault Detection and Control Systems #Neural Networks and Applications #Probability (math.PR) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2409.09498

openalex publication_date 2024/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Lévy stable motion. Otherwise, the normalized sample mean will be asymptotically normal.

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