Central Limit Theorem for the Log-Regression Wavelet Estimation of the Memory Parameter in the Gaussian Semi-Parametric Context

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator [Formula: see text]. We show that the deviation [Formula: see text], adequately normalized, is asymptotically normal and specify the asymptotic variance.