Geodesic
About conditions of gaussian approximation of kernel estimates for distribution density
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 766-776

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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).
Keywords: kernel density estimators, brownian motion, brownian bridge, KMT approximation, function of bounded variation. 
@article{SEMR_2015_12_a36,
     author = {A. S. Kartashov and A. I. Sakhanenko},
     title = {About conditions of gaussian approximation of kernel estimates for distribution density},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {766--776},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a36/}
}
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A. S. Kartashov; A. I. Sakhanenko. About conditions of gaussian approximation of kernel estimates for distribution density. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 766-776. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a36/