Geodesic
A~quadratic error of the estimation of multidimensional normal distribution densities
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 359-361

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It is proved that the distributions of the variables $$ n\int_{R^N}[P(x)-P^*_n(x)]^2dx $$ (where $P(x)$ is the density of an $N$-dimensional normal distribution, $P^*(x)$ is the corresponding empirical density, i.e. a normal density with the mean and covariance matrix equalled the empirical mean and empirical covariance matrix respectively, constructed by the sample of size $n$, $R^N$ being the $N$-dimensional space of real vectors $x=(x_1,x_2,\dots,x_N)$) converge to the distribution of the sum of two independent quadratic forms. 
@article{TVP_1968_13_2_a17,
     author = {G. M. Maniya},
     title = {A~quadratic error of the estimation of multidimensional normal distribution densities},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {359--361},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a17/}
}
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G. M. Maniya. A~quadratic error of the estimation of multidimensional normal distribution densities. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 359-361. http://geodesic.mathdoc.fr/item/TVP_1968_13_2_a17/