Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 218 / 2 / c / Solution

Let $N_{\delta }$ count sample points in the intersection of $[0,1{]}^d$ with the ${\ell }^{\infty }$ ball of radius $\delta$ around $x$. That intersection has volume at least ${\delta }^d$, so $N_{\delta }\sim \mathrm{Bin}(n,q)$ with $q\ge M{\delta }^d$. The event $\Vert X_{(L)}-x{\Vert }_{\infty }>\delta$ implies $N_{\delta }<L$. Since $\mathrm{Var}(N_{\delta })\le nq$ and $nq\ge nM{\delta }^d\ge L$, Chebyshev inequality givesTaking the minimum with the trivial bound one proves the result.

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