Asymptotic normality of wavelet estimators of the memory parameter for linear processes

We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample X-1,..., X-n of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size n -> infinity and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the scalogram for linear processes, conveniently centred and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. 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 -> infinity.