Geodesic
On sufficient conditions for a Gaussian approximation of kernel estimates for distribution densities
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1530-1552.

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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.
Keywords: kernel density estimators, brownian motion, function of bounded variation. 
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A. S. Kartashov; A. I. Sakhanenko. On sufficient conditions  for a Gaussian approximation of kernel estimates  for distribution densities. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1530-1552. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a40/

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