The squared error of the estimate of a multidimensional normal density function by the sample data

It has been proved in [2] that the distributions of the variables
$$ n\int_{R^k}[p(x)-p^*_n(x)]^2\,dx $$
converge as $n\to\infty$ to the distribution of the sum of two independent quadratic forms. In the paper the distributions of these quadratic forms are obtained.