Root-n consistent density estimators of convolutions in weighted L1L1-norms

It is known that the convolution of a smooth density with itself can be estimated at the root-n   rate by a convolution of an appropriate density estimator with itself. We show that this remains true even for discontinuous densities as long as they are of bounded variation. The assumption of bounded variation can be relaxed. We consider convergence in weighted L1L1-norms and show that the standardized convolution estimator converges in distribution to a centered Gaussian process.