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

Serge Cohen (1) , Alexander Lindner (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Institut für Mathematische Stochastik, Technische Universität Braunschweig

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.

Mots clés: Bartlett's formula; continuous time moving average process; estimation of the Hurst index; fractional Lévy process; Lévy process; limit theorem; sample autocorrelation; sample autocovariance; sample mean

Domaines: Probabilités [math.PR]

Serge Cohen, Alexander Lindner. A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process. Journal of Statistical Planning and Inference, 2013, 143 (8), pp.1295-1306. ⟨10.1016/j.jspi.2013.03.022⟩. ⟨hal-00708264⟩