In this paper we provide a theoretical contribution to the point-wise mean squared error of an adaptive multidimensional term-by-term thresholding wavelet estimator. A general result exhibiting fast rates of convergence under mild assumptions on the model is proved. It can be applied for a wide range of nonparametric models including possible dependent observations. We give applications of this result for the nonpara-metric regression function estimation problem (with random design) and the conditional density estimation problem.