Asymptotics for transportation cost in high dimensions

Abstract

LetX ₁,...,X _n ,Y ₁,...,Y _n be i.i.d. with the law μ on the cube [0, 1]^d,d⩾3. LetL _n (μ)=inf_πΣ ⁿ _i=1 ||X _i −Y _π(i)|| denote the optimal bipartite matching of theX andY points, where π ranges over all permutations of the integers 1, 2,...,n, and where ‖·‖ is a norm on ℝ^d. If μ is Lebesgue measure it is shown that

$$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha {\text{a}}{\text{.s}}{\text{.}}$$

where α is a finite constant depending on ‖ ‖ andd only. More generally, for arbitrary μ it is shown that

$$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha \int {(f{\text{(}}x{\text{)}})^{(d - 1)/d} dxa.s.} $$

wheref is the density of the absolutely continuous part of μ. We also find the rate of convergence.