On the rate of convergence of the distribution of a quadratic measure of deviation of nonparametric density estimates

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.