*(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

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 $\mathscr{H}(\beta ,L)$ 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.