Evaluation for convergence of wavelet-based estimators on fractional Brownian motion

Two wavelet-based estimators on fractional Brownian motion (FBM) are evaluated through the large deviation principle (LDP). These are σˆ_j² and Hˆ, the estimators of (i) the variance of wavelet coefficients of FBM for each scale j and (ii) the Hurst parameter, respectively, where Hˆ is obtained from the slope of the linear regression of σˆ_j² for a number of scales. Both estimators are shown to be consistent from the ergodic theorem. We perform detailed calculations related to LDP for stationary Gaussian processes with unbounded and non-L² power spectrum, to obtain L¹-estimates of the convergence of both estimators. A wavelet-based representation of the bias of the estimators is introduced and successfully used in the theory, reflecting the quantitative analysis results on FBM to the corresponding analysis of wavelet coefficients