An exponential estimate for a wavelet density estimator

This article is dedicated to deriving an exponential inequality for the distribution of the $L^p$-norm of the discrepancy between a one-dimensional probability density and its wavelet estimator that uses thresholding. In the underlying multiresolution analysis, the scale function and the mother wavelet are supposed to have compact support. The exponential estimate obtained is akin to Bernstein's inequality for sums of independent random variables. It supplements the known bounds for the mean integrated risks. The proof exploits the near-independence of empirical approximations to the coefficients of the same multiresolution level that correspond to wavelets with well-separated supports.