Journal of Mathematical Sciences

, Volume 238, Issue 4, pp 495–522 | Cite as

On Optimal Matching of Gaussian Samples

  • M. LedouxEmail author

Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let \( {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} \) be the associated empirical measure. We show that

\( \frac{1}{C}\frac{\log n}{n}\le \) 𝔼 \( \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} \)

for some numerical constant C > 0, where W2 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.


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Authors and Affiliations

  1. 1.Université de Toulouse–Paul-SabatierToulouseFrance
  2. 2.Institut Universitaire de FranceParisFrance

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