Statistica Sinica 28 (2018), 1633-1650

LIMIT BEHAVIOUR OF THE TRUNCATED PATHWISE

FOURIER-TRANSFORMATION OF LÉVY-DRIVEN

CARMA PROCESSES FOR NON-EQUIDISTANT

DISCRETE TIME OBSERVATIONS

Żywilla Fechner and Robert Stelzer

University of Silesia in Katowice and Ulm University

Abstract: This paper considers a continuous time analogue of the classical autoregressive moving average processes, Lévy-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.

Key words and phrases: CARMA process, frequency domain, high-frequency data, Lévy process, trapezoidal rule.