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

Abstract : 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 $X_1,\dots,X_n$ 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 $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$.
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https://hal.archives-ouvertes.fr/hal-00277645
Contributor : François Roueff Connect in order to contact the contributor
Submitted on : Tuesday, May 6, 2008 - 7:25:35 PM
Last modification on : Monday, October 19, 2020 - 9:49:26 AM
Long-term archiving on: : Friday, May 28, 2010 - 6:37:29 PM

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### Citation

François Roueff, Murad Taqqu. Asymptotic normality of wavelet estimators of the memory parameter for linear processes. Journal of Time Series Analysis, Wiley-Blackwell, 2009, 30 (5), pp.Pages 467-575. ⟨10.1111/j.1467-9892.2009.00627.x⟩. ⟨hal-00277645⟩

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