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Zbl 1150.62058
Moulines, E.; Roueff, F.; Taqqu, M.S.
On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter.
(English)
[J] J. Time Ser. Anal. 28, No. 2, 155-187 (2007). ISSN 0143-9782; ISSN 1467-9892/e

A process $X$ is said to have memory parameter $d\in R$ ($M(d)$ process), if for any integer $K>d-1/2$, the $K$th-order difference process $\Delta\sp{K}X$ is weakly stationary with spectral density function $$f\sb{\Delta\sp{K}X}(\lambda)=\vert 1-e\sp{-i\lambda}\vert\sp{2(K-d)}f\sp{*}(\lambda),\quad \lambda\in(-\pi,\pi),$$ where $f\sp{*}$ is some non-negative symmetric function. The authors derive an explicit expression for the covariance and spectral density of the wavelet coefficients of an $M(d)$ process at a given scale. If $f\sp{*}$ belongs to a class of smooth functions ${\Cal H}(\beta,L)$ it is shown that the spectral density of the wavelet coefficients of an $M(d)$ process can be approximated, at large scales, by the spectral density of the wavelet coefficients of a fractional Brownian motion.\par An explicit bound for the difference between these two quantities is derived. The authors show that the relative $L\sp{\infty}$ error between the spectral densities of the wavelet coefficients decreases exponentially fast to zero with a rate given by the smoothness exponent $\beta$ of $f\sp{*}$. Gaussian processes are considered and an explicit expression for the limiting variance of the estimator of $d$ based on the regression of the log-scale spectrum is obtained.
[Aleksandr D. Borisenko (Ky{\"\i}v)]
MSC 2000:
*62M15 Spectral analysis of processes
42C40 Wavelets
62M10 Time series, etc. (statistics)

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