= Pointwise central limit theorem for a kernel density estimator
{title2=$\sqrt{nh}(\widehat f_{n,h}(x)-f(x))\xrightarrow{d}N(0,f(x)\|K\|_2^2)$}
Let the <kernel for density estimation> have integral one, be bounded and have compact support. For a <probability density function> with bounded derivative, assume $h\to0$, $nh\to\infty$ and $nh^3\to0$. The <bias of a kernel density estimator> is $O(h)$, negligible on the $\sqrt{nh}$ scale. For $Y_{n,i}=K((x-X_i)/h)$, $h^{-1}\operatorname{Var}Y_{n,i}\to f(x)\|K\|_2^2$. The centered summands divided by $\sqrt{nh}$ are uniformly small, so the <Lindeberg-Feller central limit theorem> applies. The condition $nh^3\to0$ 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.
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