A quadratic error of the estimation of multidimensional normal distribution densities

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.