A process is said to have memory parameter ( process), if for any integer , the th-order difference process is weakly stationary with spectral density function
where is some non-negative symmetric function. The authors derive an explicit expression for the covariance and spectral density of the wavelet coefficients of an process at a given scale. If belongs to a class of smooth functions it is shown that the spectral density of the wavelet coefficients of an 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 error between the spectral densities of the wavelet coefficients decreases exponentially fast to zero with a rate given by the smoothness exponent of . Gaussian processes are considered and an explicit expression for the limiting variance of the estimator of based on the regression of the log-scale spectrum is obtained.