Estimates of the rate of approximation in the CLT for $L_1$-norm of   density estimators
  We provide estimates of the rate of strong approximation and bounds for probabilities of moderate deviations in the CLT for the $L_1$-norm of the kernel density estimator without any assumptions on the density and assuming that the kernel is bounded and has bounded support. 
Andrei Yu. Zaitsev
math.PR