Pointwise central limit theorem for a kernel density estimator ($\sqrt{nh}({\hat{f}}_{n,h}(x)-f(x))\stackrel{d}{\to }N(0,f(x)\Vert K{\Vert }_2^2)$)

Let the kernel for density estimation have integral one, be bounded and have compact support. For a probability density function with bounded derivative, assume $h\to 0$, $nh\to \infty$ and $n{h}^3\to 0$. The bias of a kernel density estimator is $O(h)$, negligible on the $\sqrt{nh}$ scale. For $Y_{n,i}=K((x-X_i)/h)$, $h^{-1}\mathrm{Var}{Y}_{n,i}\to f(x)\Vert K{\Vert }_2^2$. The centered summands divided by $\sqrt{nh}$ are uniformly small, so the Lindeberg-Feller central limit theorem applies. The condition $n{h}^3\to 0$ alone is insufficient: with $h=n^{-2}$ and a compactly supported kernel, with probability tending to one no observation enters the kernel window, producing a degenerate limit at a point with positive density.