Limit behaviour of the truncated pathwise Fourier-transformation of L\'evy-driven CARMA processes for non-equidistant discrete time observations

TL;DR

This paper investigates the limiting behavior of the truncated Fourier transform of Lévy-driven CARMA processes, analyzing both continuous observations and non-equidistant high-frequency data, with results on convergence and asymptotic normality.

Contribution

It provides new theoretical insights into the limit properties of the truncated Fourier transform for Lévy-driven CARMA processes, including convergence and asymptotic normality under high-frequency sampling.

Findings

Limiting properties of the periodogram match discrete-time cases.

Convergence of numerical approximation under certain sampling conditions.

Asymptotic normality of the Fourier transform estimator.

Abstract

This paper considers a continuous time analogue of the classical autoregressive moving average processes, L\'evy-driven CARMA processes. First we describe limiting properties of the periodogram by means of the so-called truncated Fourier transform if observations are available continuously. The obtained results are in accordance with their counterparts from the discrete-time case. Then we discuss the numerical approximation of the truncated Fourier transform based on non-equidistant high frequency data. In order to ensure convergence of the numerical approximation to the true value of the truncated Fourier transform a certain control on the maximal distance between observations and the number of observations is needed. We obtain both convergence to the continuous time quantity and asymptotic normality under a high-frequency infinite time horizon limit.