Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 27K / d / Solution

Fix an integer $k>n$ and partition $[0,1]$, up to endpoints of measure zero, intoFor $p\ne q$, defineThese functions have integral zero and $L^2$ norm one, so they are admissible.

Letbe the interval occupancy counts. Since each $Z_i$ is uniform, $\mathbb{E}{N}_p=n/k$, and henceFor every outcome,because the nonnegative integer counts satisfy $N_p^2\ge N_p$ and ${\sum }_p{N}_p=n$. No independence assumption was used.

Averaging over all ordered pairs givesThe diagonal terms vanish, so some admissible pair $p\ne q$ satisfies the same lower bound. ThereforeLetting $k\to \infty$ proves the dependent uniform sampling variance lower bound