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

Abstract

Two wavelet-based estimators on fractional Brownian motion (FBM) are evaluated through the large deviation principle (LDP). These are /spl sigma//spl circ//sub j//sup 2/ and H/spl circ/, the estimators of (i) the variance of wavelet coefficients of FBM for each scale j and (ii) the Hurst parameter, respectively, where H/spl circ/ is obtained from the slope of the linear regression of /spl sigma//spl circ//sub j//sup 2/ 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-L/sup 2/ power spectrum, to obtain L/sup 1/-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.