## Electronic Journal of Statistics

### Noise recovery for Lévy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums

#### Abstract

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 high-frequency 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. [11] is given.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 533-561.

Dates
First available in Project Euclid: 6 March 2013

https://projecteuclid.org/euclid.ejs/1362579369

Digital Object Identifier
doi:10.1214/13-EJS783

Mathematical Reviews number (MathSciNet)
MR3035265

Zentralblatt MATH identifier
1337.62260

#### Citation

Ferrazzano, Vincenzo; Fuchs, Florian. Noise recovery for Lévy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums. Electron. J. Statist. 7 (2013), 533--561. doi:10.1214/13-EJS783. https://projecteuclid.org/euclid.ejs/1362579369

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