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Asymptotic normality of wavelet estimators of the memory parameter for linear processes. (English) Zbl 1224.62068
The authors introduce a new class M$(d)$ of real-valued processes and investigate the properties of these processes. The process $X =\{X_k\}_{k\in\Bbb Z}$ is an M$(d)$ process if it has the memory parameter $d$, $d\in\Bbb R$, and the short-range spectral density $f^*(\lambda)$. That is, for any integer $k >d-1/2$, the $k$-th order difference process $\Delta^kX$ is weakly stationary with spectral density function $$f_{\Delta^kX}(\lambda)=\vert 1-e^{-i\lambda}\vert ^{2(k-d)} f^*(\lambda),$$ where $f^*(\lambda)$ is a non-negative symmetric function which is continuous and nonzero at the origin. The class M$(d)$ includes both stationary and nonstationary processes, depending on the value of the memory parameter $d$. The function $f(\lambda)=\vert 1-e^{-i\lambda}\vert ^{-2d} f^*(\lambda)$ is called the generalized spectral density of $X$. It is a proper spectral density function when $d<1/2$. In this case the process $X$ is covariance stationary with spectral density function $f(\lambda)$. The process $X$ is said to have long memory if $0< d < 1/2$, short memory if $d=0$ and negative memory if $d < 0$. The process is not invertible if $d<-1/2$. The factor $f^*(\lambda)$ is a nuisance function which determines the `short-range’ dependence. The authors propose semi-parametric estimates for the memory parameter $d$ using wavelets from a sample $X_1,\dots,X_n$ of the process. Two estimators of the memory parameter $d$ are considered: the log-regression wavelet estimator and the wavelet Whittle estimator. It is shown that these estimators are asymptotically normal as the sample size $n\to\infty$. An explicit expression for the limit variance is obtained. To study the asymptotic properties of these estimators the authors use a central limit theorem for scalograms, that is arrays of quadratic forms of the observed sample computed from the wavelet coefficients of this sample. For more details see {\it F. Roueff} and {\it M.S. Taqqu} [Stochastic Processes Appl. 119, No. 9, 3006--3041 (2009; Zbl 1173.60311)]. In contrast to quadratic forms computed on the basis of Fourier coefficients, such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$. This allows extending to the non-Gaussian linear process settings asymptotic normality results obtained for Gaussian processes. See {\it E. Moulines, F. Roueff} and {\it M.S. Taqqu}, Fractals 15, No. 4, 301--313 (2007; Zbl 1141.62073), and {\it E. Moulines, F. Roueff} and {\it M.S. Taqqu} [Ann. Stat. 36, No. 4, 1925-1956 (2008; Zbl 1142.62062).

62M10Time series, auto-correlation, regression, etc. (statistics)
62M15Spectral analysis of processes
42C40Wavelets and other special systems
62G05Nonparametric estimation
60F05Central limit and other weak theorems
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