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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) Zbl 1150.62058

A process $X$ is said to have memory parameter $d\in R$ ($M\left(d\right)$ process), if for any integer $K>d-1/2$, the $K$th-order difference process ${{\Delta }}^{K}X$ is weakly stationary with spectral density function

${f}_{{{\Delta }}^{K}X}\left(\lambda \right)={|1-{e}^{-i\lambda }|}^{2\left(K-d\right)}{f}^{*}\left(\lambda \right),\phantom{\rule{1.em}{0ex}}\lambda \in \left(-\pi ,\pi \right),$

where ${f}^{*}$ is some non-negative symmetric function. The authors derive an explicit expression for the covariance and spectral density of the wavelet coefficients of an $M\left(d\right)$ process at a given scale. If ${f}^{*}$ belongs to a class of smooth functions $ℋ\left(\beta ,L\right)$ it is shown that the spectral density of the wavelet coefficients of an $M\left(d\right)$ process can be approximated, at large scales, by the spectral density of the wavelet coefficients of a fractional Brownian motion.

An explicit bound for the difference between these two quantities is derived. The authors show that the relative ${L}^{\infty }$ error between the spectral densities of the wavelet coefficients decreases exponentially fast to zero with a rate given by the smoothness exponent $\beta$ of ${f}^{*}$. 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.

##### MSC:
 62M15 Spectral analysis of processes 42C40 Wavelets and other special systems 62M10 Time series, auto-correlation, regression, etc. (statistics)