Interest in continuous-time processes has increased rapidly in recent years, largely because of high-frequency data available in many applications. We develop a method for estimating the kernel function g of a second-order stationary Lévy-driven continuous-time moving average (CMA) process Y based on observations of the discrete-time process Y ∆ obtained by sampling Y at ∆, 2∆, . . . , n∆ for small ∆. We approximate g by g ∆ based on the Wold representation and prove its pointwise convergence to g as ∆ → 0 for CARMA(p, q) processes. Two non-parametric estimators of g ∆ , based on the innovations algorithm and the Durbin-Levinson algorithm, are proposed to estimate g. For a Gaussian CARMA process we give conditions on the sample size n and the grid-spacing ∆(n) under which the innovations estimator is consistent and asymptotically normal as n → ∞. The estimators can be calculated from sampled observations of any CMA process and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally we extend results of Brockwell et al. (2012) for sampled CARMA processes to a much wider class of CMA processes.

Continuous-time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non-uniformly spaced data and as a tool for dealing with high-frequency data of the form Y nD ,n ¼ 0, 1, 2,…, where D is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process ðY nD Þ n‰Z , when D is small and the underlying continuous-time process ðY t Þ t‰R is a specified CARMA process.

We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean Lévy processes. An L 2 -consistent estimator for the increments of the driving Lévy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the highfrequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average models on a discrete grid. We compare their autocovariance structure with the one of sampled CARMA processes and show that the rule of integration plays a crucial role. Moreover, new insight into the kernel estimation procedure of Brockwell et al. (2012a) is given.

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