Comptes Rendus
Probability theory
Scaling and non-standard matching theorems
[Mise à l'échelle et théorèmes d'appariement non-standard]
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 692-695.

Consider the standard Gaussian measure μ on R2. Consider independent r.v.s (Xi)iN distributed according to μ, and an independent copy (Yi)iN of these r.v.s. We prove that, for some number C and N large, we have

(logN)2CEinfπiNd(Xi,Yπ(i))2C(logN)2,(1)
where the infimum is over all permutations π of {1,,N}. The striking point of this result is the factor (logN)2. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is logN. The upper bound was proved by Michel Ledoux (2017) [3].

Considérons une suite indépendente (Xi)iN de variables aléatoires distribuées comme la mesure gaussienne canonique μ sur R2 et une copie independente (Yi)iN de cette même suite. Pour une certaine constante universelle C et N2, nous avons les inégalités

(logN)2CEinfπiNd(Xi,Yπ(i))2C(logN)2(1)
où l'infimum est pris sur toutes les permutations π de {1,,N}. La borne supérieure a été prouvée par Michel Ledoux (2017) [3], qui conjecturait que l'inégalité (1) était correcte avec un facteur logN et non pas (logN)2. C'est précisement l'apparence de ce facteur (logN)2 qui est non standard.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.018

Michel Talagrand 1

1 23, rue Louis-Pouey, 92800 Puteaux, France
@article{CRMATH_2018__356_6_692_0,
     author = {Michel Talagrand},
     title = {Scaling and non-standard matching theorems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {692--695},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.018},
     language = {en},
}
TY  - JOUR
AU  - Michel Talagrand
TI  - Scaling and non-standard matching theorems
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 692
EP  - 695
VL  - 356
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2018.04.018
LA  - en
ID  - CRMATH_2018__356_6_692_0
ER  - 
%0 Journal Article
%A Michel Talagrand
%T Scaling and non-standard matching theorems
%J Comptes Rendus. Mathématique
%D 2018
%P 692-695
%V 356
%N 6
%I Elsevier
%R 10.1016/j.crma.2018.04.018
%G en
%F CRMATH_2018__356_6_692_0
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/

[1] M. Ajtai; J. Komlós; G. Tusnády On optimal matchings, Combinatorica, Volume 4 (1984) no. 4, pp. 259-264

[2] L. Ambrosio; F. Stra; D. Trevisan A PDE approach to a 2-dimensional matching problem, Probab. Theory Relat. Fields (2016) (in press)

[3] M. Ledoux On optimal matching of Gaussian samples, Zap. Nauč. Semin. POMI, Volume 457 (2017) (Veroyatnost' i Statistika 25 226–264)

[4] M. Talagrand Upper and Lower Bounds for Stochastic Processes http://michel.talagrand.net/ULB.pdf (new edition in preparation, available at)

[5] J. Yukich Some generalizations of the Euclidean two-sample matching problem, Probability in Banach Spaces, 8, Progress in Probability, vol. 30, Birkhäuser, 1992, pp. 55-66

  • Eustasio del Barrio; Alberto González-Sanz; Jean-Michel Loubes Central limit theorems for general transportation costs, Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, Volume 60 (2024) no. 2, pp. 847-873 | DOI:10.1214/22-aihp1356 | Zbl:1546.49083
  • Emanuele Caglioti; Francesca Pieroni Random matching in 2D with exponent 2 for Gaussian densities, Journal of Statistical Physics, Volume 191 (2024) no. 5, p. 34 (Id/No 62) | DOI:10.1007/s10955-024-03275-y | Zbl:1542.60020
  • Michel Ledoux Optimal matching of random samples and rates of convergence of empirical measures, Mathematics going forward. Collected mathematical brushstrokes, Cham: Springer, 2023, pp. 615-627 | DOI:10.1007/978-3-031-12244-6_43 | Zbl:1536.60015
  • Mario Ghossoub; Jesse Hall; David Saunders Maximum Spectral Measures of Risk with Given Risk Factor Marginal Distributions, Mathematics of Operations Research, Volume 48 (2023) no. 2, p. 1158 | DOI:10.1287/moor.2022.1299
  • Martin Huesmann; Francesco Mattesini; Dario Trevisan Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus, Stochastic Processes and their Applications, Volume 155 (2023), pp. 1-26 | DOI:10.1016/j.spa.2022.09.008 | Zbl:1508.60050
  • Luigi Ambrosio; Michael Goldman; Dario Trevisan On the quadratic random matching problem in two-dimensional domains, Electronic Journal of Probability, Volume 27 (2022), p. 35 (Id/No 54) | DOI:10.1214/22-ejp784 | Zbl:1487.60017
  • Michel Ledoux; Jie-Xiang Zhu On optimal matching of Gaussian samples. III, Probability and Mathematical Statistics, Volume 41 (2021) no. 2, pp. 237-265 | DOI:10.37190/0208-4147.41.2.3 | Zbl:1482.60013
  • Łukasz Szpruch; Alvin Tse Antithetic multilevel sampling method for nonlinear functionals of measure, The Annals of Applied Probability, Volume 31 (2021) no. 3, pp. 1100-1139 | DOI:10.1214/20-aap1614 | Zbl:1479.60005
  • Michel Talagrand Matching Theorems, Upper and Lower Bounds for Stochastic Processes, Volume 60 (2021), p. 111 | DOI:10.1007/978-3-030-82595-9_4
  • Martin Kenyeres; Jozef Kenyeres, 2020 IEEE 18th World Symposium on Applied Machine Intelligence and Informatics (SAMI) (2020), p. 157 | DOI:10.1109/sami48414.2020.9108754
  • Sergio Caracciolo; Matteo P. D'Achille; Vittorio Erba; Andrea Sportiello The Dyck bound in the concave 1-dimensional random assignment model, Journal of Physics A: Mathematical and Theoretical, Volume 53 (2020) no. 6, p. 25 (Id/No 064001) | DOI:10.1088/1751-8121/ab4a34 | Zbl:1511.82019
  • Dario Benedetto; Emanuele Caglioti Euclidean random matching in 2D for non-constant densities, Journal of Statistical Physics, Volume 181 (2020) no. 3, pp. 854-869 | DOI:10.1007/s10955-020-02608-x | Zbl:1458.60020
  • Luigi Ambrosio; Federico Glaudo Finer estimates on the 2-dimensional matching problem, Journal de l'École Polytechnique – Mathématiques, Volume 6 (2019), pp. 737-765 | DOI:10.5802/jep.105 | Zbl:1434.60054
  • Sergio Caracciolo; Matteo D'Achille; Gabriele Sicuro Anomalous scaling of the optimal cost in the one-dimensional random assignment problem, Journal of Statistical Physics, Volume 174 (2019) no. 4, pp. 846-864 | DOI:10.1007/s10955-018-2212-9 | Zbl:1451.90137

Cité par 14 documents. Sources : Crossref, zbMATH

Commentaires - Politique