= Integrated variance of a kernel density estimator
{title2=$\mathbb E\|\widehat f_{n,h}-K_h*f\|_2^2$}
For an $L^2$ kernel and an independent sample from a density $f$, the centred <kernel density estimator> has
$$
\mathbb E\|\widehat f_{n,h}-K_h*f\|_2^2=\frac1n\left(\frac{\|K\|_2^2}{h}-\|K_h*f\|_2^2\right).
$$
Independence removes cross terms and <Tonelli theorem> integrates the pointwise variance. <Young's convolution inequality> ensures $K_h*f\in L^2$ even without $f\in L^2$. The <Cauchy-Schwarz inequality> yields the expected-norm bound $\mathbb E\|\widehat f_{n,h}-K_h*f\|_2\le\|K\|_2/\sqrt{nh}$.
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