The Integrability of the Square Exponential Transportation Cost

Abstract

Let $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ be i.i.d. with the uniform distribution on $(\lbrack 0,1\rbrack^2, \| \|)$, where $\| \|$ denotes the Euclidean norm. Using a new presentation of the Ajtai-Komlos-Tusnady (AKT) transportation algorithm, it is shown that the square exponential transportation cost $\inf_\pi \sum^n_{i=1} \exp\Bigg(\frac{\|X_i - Y_{\pi(i)}\|}{K(\log n/n)^{1/2}}\Bigg)^2,$ where $\pi$ ranges over all permutations of the integers $1,\ldots,n$, satisfies an integrability condition. This condition strengthens the optimal matching results of AKT and supports a recent conjecture of Talagrand. Rates of growth for the $L_p$ transportation cost are also found.