L p - wavelet regression with correlated errors and inverse problems

We investigate global performances of non-linear wavelet estimation in regression models with correlated errors. Convergence properties are studied over a wide range of Besov classes B π,r and for a variety of L error measures. We consider error distributions with Long-Range-Dependence parameter α, 0 < α ≤ 1. In this setting we present a single adaptive wavelet thresholding estimator which achieves near-optimal properties simultaneously over a class of spaces and error measures. Our method reveals an elbow feature in the rate of convergence at s = α 2 ( p π − 1) when p > 2 α +π. Using a vaguelette decomposition of fractional Gaussian noise we draw a parallel with certain inverse problems where similar rate results occur.