On the sample autocovariance of a L\'evy driven moving average process when sampled at a renewal sequence

TL;DR

This paper investigates the statistical properties of a Lévy driven moving average process sampled at renewal times, establishing asymptotic normality for key estimators and applying results to parameter estimation of Lévy-driven Ornstein-Uhlenbeck processes.

Contribution

It introduces asymptotic normality results for sample autocovariance and autocorrelation of Lévy driven processes sampled at renewal times, extending classical sampling results.

Findings

Asymptotic normality of sample mean, autocovariance, and autocorrelation established.

Comparison between renewal sampling and equidistant sampling methods.

Application to parameter estimation of Lévy-driven Ornstein-Uhlenbeck process.

Abstract

We consider a L\'evy driven continuous time moving average process $X$ sampled at random times which follow a renewal structure independent of $X$. Asymptotic normality of the sample mean, the sample autocovariance, and the sample autocorrelation is established under certain conditions on the kernel and the random times. We compare our results to a classical non-random equidistant sampling method and give an application to parameter estimation of the L\'evy driven Ornstein-Uhlenbeck process.