Geodesic
On the rate of convergence of the distribution of a~quadratic measure of deviation of nonparametric density estimates
Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 164-170

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Let $X_1, X_2, \dots$ be a sequence of independent identically distributed real-valued random variables with density $f(x)$ and let $f_n(x)$ he a Parzen's estimate of $f(x)$. We prove that the distribution of a quadratic functional $$ \int_{-\infty}^\infty[f_n(x)-f(x)]^2a(x)\,dx $$ is asymptotically normal and obtain some estimates of the rate of convergence. 
@article{TVP_1984_29_1_a21,
     author = {\v{S}. A. Ha\v{s}imov},
     title = {On the rate of convergence of the distribution of a~quadratic measure of deviation of nonparametric density estimates},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {164--170},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a21/}
}
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Š. A. Hašimov. On the rate of convergence of the distribution of a~quadratic measure of deviation of nonparametric density estimates. Teoriâ veroâtnostej i ee primeneniâ, Tome 29 (1984) no. 1, pp. 164-170. http://geodesic.mathdoc.fr/item/TVP_1984_29_1_a21/