On sufficient conditions for a Gaussian approximation of kernel estimates for distribution densities

Recently E. Gine, V. Koltchinskii and L. Sakhanenko (Ann. Probab., 2004) investigated necessary and sufficient conditions for weak convergence to the double exponential distribution of a normalized random variable $ \sup\nolimits_{t \in \mathbb{R}} \left | \psi(t) (f_n(t) - \mathbf{E} f_n (t)) \right | $ with some weight function $\psi(t)$, where $f_n$ is a kernel density estimator. The proof of their results consists of a large number of technically difficult stages and uses more than fifteen bulky assumptions. In this work we prove that sufficiency of convergence can be obtained under simpler and wider assumptions.