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Central limit theorem for the log-regression wavelet estimation of the memory parameter in the Gaussian semi-parametric context. (English) Zbl 1141.62073
Summary: We consider a Gaussian time series, stationary or not, with long memory exponent $d\in\Bbb R$. The generalized spectral density function of the time series is characterized by $d$ and by a function $f^*(\lambda)$ 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 $\lambda = 0$ 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 $d\ne 0$. We establish a central limit theorem for the resulting estimator $\widehat d$. We show that the deviation $\widehat d-d$, adequately normalized, is asymptotically normal and specify the asymptotic variance.

MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 60F05 Central limit and other weak theorems 62G08 Nonparametric regression 62M15 Spectral analysis of processes 42C40 Wavelets and other special systems 60G15 Gaussian processes
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