Wavelet coefficients of a process have arguments shift and scale. It can thus be viewed as a time series along shift for each scale. We have considered in the previous study general wavelet coefficient domain estimators and revealed a localization property with respect to shift. In this paper, we formulate the localization property with respect to scale, which is more difficult than that of shift. Two factors that govern the decay rate of cross-scale covariance are indicated. The factors are both functions of vanishing moments and scale-lags. The localization property is then successfully applied to formulate limiting variance in the central limit theorem associated with Hurst index estimation problem of fractional Brownian motion. Especially, we can find the optimal upper bound $J$ of scales $1, \ldots, J$ used in the estimation to be $J = 5$ by an evaluation of the diagonal component of the limiting variance, in virtue of the scale localization property.
"On a localization property of wavelet coefficients for processes with stationary increments, and applications. II. Localization with respect to scale." Osaka J. Math. 51 (1) 1 - 39, January 2014.