Matching Random Samples in Many Dimensions

Abstract

Consider any norm $N$ on $\mathbb{R}^d, d \geq 3$, and independent uniformly distributed points $X_1, \ldots, X_n, \ldots; Y_1, \ldots, Y_n, \ldots$ in $\lbrack 0, 1\rbrack^d$. Consider the random variable $M_n = \inf \sum_{i \leq n} N(X_i - Y_{\sigma(i)})$, where the infimum is taken over all permutations $\sigma$ of $\{1, \ldots, n\}$. We show that for some universal constant $K$, we have $\lim \sup_{n \rightarrow \infty} M_n n^{-1 + 1/d} \leq r_N \big(1 + K \frac{\log d}{d}\big)mathrm{a,s.},$ where $r_N$ is the radius of the ball for $N$ of volume 1.