Estimates for functionals of a density constructed from discretized observations

We consider the problem of estimating a functional $G(f)=\int_Kg(f(x))\,dx$ of an unknown density $f(x)$ of a distribution concentrated on an $r$-dimensional unit cube $K$, where $g$ is a sufficiently smooth function, on the basis of the discretization of the independent observations $X_1,\dots,X_n$ with density $f(x)$. We give an estimate for $G_n$ that is constructed on the basis of the discretization of $n$ observations with step $1/n$ and give conditions under which the variable $\gamma_n=n^{1/2}(G_n-G(f))$ is asymptotically normal as $n\to\infty$. In the case when $r=1$ the limit variance is minimal.