ON OPTIMAL MATCHING OF GAUSSIAN SAMPLES

摘要

Let X_1,..., X_n be independent random variables having as common distribution the standard Gaussian measure u on R~2 and let μ_n = 1 Σ_(i=1)~n δ_(X_i) be the associated empirical measure. We show that 1/C (log n)≤E(W_2~2(μ_n,μ)) ≤ C((log n)~2) for some numerical constant C > 0, where W_2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan. Bibliography: 39 titles.