About conditions of gaussian approximation of kernel estimates for distribution density

Recently E. Gine, V. Koltchinskii and L. Sakhanenko (2004) investigated asymptotical behavior of a random variable of the form $\sqrt{n h_n} \sup\nolimits_{t \in \mathbf{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 limit theorems consists of a large number of technically difficult stages and uses more than ten bulky assumptions. In this work we show that under simpler and wider conditions the above stated problem is reduced to the study of asymptotics of a supremum of some special Gaussian process. The obtained result can be used in further investigation of functionals based on empirical processes and kernel density estimators. Our proof is based on the well-known approximation of Komlos, Major and Tusnady (1975).