Scaling and non-standard matching theorems

Michel Talagrand

doi:10.1016/j.crma.2018.04.018

Probability theory

Scaling and non-standard matching theorems
[Mise à l'échelle et théorèmes d'appariement non-standard]

Présenté par : Gilles Pisier

Michel Talagrand ¹

¹ 23, rue Louis-Pouey, 92800 Puteaux, France

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 692-695.

Résumé
Abstract

Considérons une suite indépendente ${(X_{i})}_{i \leq N}$ de variables aléatoires distribuées comme la mesure gaussienne canonique μ sur $R^{2}$ et une copie independente ${(Y_{i})}_{i \leq N}$ de cette même suite. Pour une certaine constante universelle C et $N \geq 2$ , nous avons les inégalités

\frac{{(\log N)}^{2}}{C} \leq E \inf_{π} \sum_{i \leq N} d {(X_{i}, Y_{π (i)})}^{2} \leq C {(\log N)}^{2}

(1)

où l'infimum est pris sur toutes les permutations π de

{1, \dots, N}

. La borne supérieure a été prouvée par Michel Ledoux (2017) [3], qui conjecturait que l'inégalité (1) était correcte avec un facteur

\log N

et non pas

{(\log N)}^{2}

. C'est précisement l'apparence de ce facteur

{(\log N)}^{2}

qui est non standard.

Consider the standard Gaussian measure μ on $R^{2}$ . Consider independent r.v.s ${(X_{i})}_{i \leq N}$ distributed according to μ, and an independent copy ${(Y_{i})}_{i \leq N}$ of these r.v.s. We prove that, for some number C and N large, we have

\frac{{(\log N)}^{2}}{C} \leq E \inf_{π} \sum_{i \leq N} d {(X_{i}, Y_{π (i)})}^{2} \leq C {(\log N)}^{2},

(1)

where the infimum is over all permutations π of

{1, \dots, N}

. The striking point of this result is the factor

{(\log N)}^{2}

. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is

\log N

. The upper bound was proved by Michel Ledoux (2017) [3].

Reçu le : 2018-04-11
Accepté le : 2018-04-13
Publié le : 2018-05-25

DOI : 10.1016/j.crma.2018.04.018

Affiliations des auteurs :

Michel Talagrand ¹

¹ 23, rue Louis-Pouey, 92800 Puteaux, France

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Michel Talagrand. Scaling and non-standard matching theorems. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 692-695. doi : 10.1016/j.crma.2018.04.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.018/

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Cité par

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