A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process

• A central limit theorem for the sample autocorrelations of Lévy driven continuous time moving average processes is obtained.
• Unlike for discrete time moving average processes with i.i.d. noise, a correction term has to be added to the classical Bartlett formula.
• Equidistantly sampled Lévy driven continuous time moving average process need not have a representation as discrete time moving average processes with i.i.d. noise.
• Asymptotically normal estimators for the Hurst index of fractional Lévy noise are obtained.

In this article we consider Lévy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional Lévy processes is also deduced from these results.