On optimal scale upper bound in wavelet-based estimation for hurst index of fractional Brownian motion

We give an evaluation of limit variance sigma(2)(H,J) in the central limit theorem for wavelet-based estimator of the Hurst index of fractional Brownian motion (FBM). sigma(2)(H,J) is the variance of a certain linear sum (with respect to scale j) of random variables. As a result, contribution to variance of the linear form turns out to be mostly its diagonal part components, which can be considered as a form of frequency localization (FL) property of wavelet coefficients of FBM. This FL property, together with specifiability of small J behavior of sigma(2)(H,J), is applied to determine the optimal upper bound of scale j used in the estimation.