Tusnády's problem and online two-sample thinning

Abstract

We determine the sharp order of discrepancy in Tusnády's problem in every fixed dimension $d\ge 2$. The largest combinatorial discrepancy of $N$ points in $\mathbb{R}^d$ with respect to axis-parallel boxes is $Θ_d((\log N)^{d-1/2})$, extending Wei's planar result and matching Nikolov's upper bound. More strongly, with high probability, this lower bound holds simultaneously for every subset obtained by deleting at most a sufficiently small fixed fraction of a uniform $N$-point sample. We apply this geometric estimate to online thinning of two independent samples of size $n$ from an unknown common distribution, where observations must be kept or discarded upon arrival. For every fixed dimension $k\ge 1$ and expected deletion budget $ρn$, with $0<ρ<2$, we prove that the minimax expected prefix discrepancy over lower orthants is $Θ_{k,ρ}((\log n)^{k+1/2})$. In dimension one, the optimal terminal discrepancy is instead $Θ_ρ((\log\log n)^{3/2})$, giving a sharp separation between the prefix and terminal objectives. For $k\ge 2$, we obtain terminal bounds differing by a factor of $\log n$. The lower bounds are based on Gaussian small-ball estimates for box-incidence matrices, while the upper bounds use stationary Metropolis thinning and, in one dimension, a terminal cancellation procedure.