= Dependent uniform sampling variance lower bound
Let $Z_1,\ldots,Z_n$ each be uniform on $[0,1]$, with arbitrary dependence. For centered unit-norm $f\in L^2[0,1]$, put
$$
D(f)=\operatorname{Var}\left(\frac1n\sum_{i=1}^nf(Z_i)\right).
$$
Then
$$
\sup_fD(f)\ge\frac1n.
$$
Partition $[0,1]$ into $k$ equal intervals and test normalized differences of two interval indicators. Averaging over the two intervals reduces the claim to $\sum_pN_p^2\ge\sum_pN_p=n$ for their occupancy counts, and then let $k\to\infty$.
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