Densité et dimension
Annales de l'Institut Fourier, Volume 33 (1983) no. 3, pp. 233-282.

A class 𝒮 of subsets of a set X is called a Vapnik-Cervonenkis class if the growth of the function Δ 𝒮 :r Sup {|A||AX,|A|=r} is polynomial ; these classes have proved to be useful in Statistics and Probability (see for example Vapnik, Cervonenkis [V.N. Vapnik, A.YA. Cervonenkis, Theor. Prob. Appl., 16 (1971), 264-280], Dudley [R.M. Dudley, Ann. of Prob., 6 (1978), 899-929]).

The present paper is a survey on Vapnik-Cervonenkis classes. Moreover we prove here many new results, among them the following:

- a subset 𝒮 of 2 X is a Vapnik-Cervonenkis class if and only if the number of atoms of the σ-algebra generated by any collection of r members of 𝒮 if no more than Cr s (where C and s are two positive real numbers);

- if 𝒮 is a subset of 2 X , every probability law P on the σ-algebra generated by 𝒮 defines a semimetric d p :S,S P(SΔS ) on the class 𝒮, and the entropy dimension of the space (𝒮,d p ) will be denoted dim ¯(𝒮,d p ) ; the class 𝒮 is a Vapnik-Cervonenkis class if and only if Sup P dim ¯(𝒮,d p ) is finite.

The present paper contains other new results, some of them being stated without proof in my note [P. Assouad, C.R.A.S., Paris, 292 (1981), 921-924]).

Une partie 𝒮 de 2 X est appelée une classe de Vapnik-Cervonenkis si la croissance de la fonction Δ 𝒮 :r Sup {|A||AX,|A|=r} est polynomiale; ces classes se trouvent être utiles en Statistique et en Calcul des Probabilités (voir par exemple Vapnik, Cervonenkis [V.N. Vapnik, A.YA. Cervonenkis, Theor. Prob. Appl., 16 (1971), 264-280], Dudley [R.M. Dudley, Ann. of Prob., 6 (1978), 899-929]).

Le présent travail est un essai de synthèse sur les classes de Vapnik-Cervonenkis. Mais il contient aussi beaucoup de résultats nouveaux, et notamment les deux résultats suivants :

- une partie 𝒮 de 2 X est une classe de Vapnik-Cervonenkis si et seulement si le nombre d’atomes de la tribu engendrée par r membres quelconques de 𝒮 est majoré par un polynôme en r ;

- si 𝒮 est une partie de 2 X , chaque loi de probabilité P sur la tribu engendrée par 𝒮 définit un écart d p :S,S P(SΔS ) sur la famille 𝒮, et on note dim(𝒮,d p ) la dimension d’entropie de l’espace (𝒮,d p ); la famille 𝒮 est une classe de Vapnik-Cervonenkis si et seulement si la quantité Sup dim ¯(𝒮,d p ) est finie.

On trouvera dans l’introduction les énoncés de plusieurs autres résultats nouveaux démontrés ici (dont certains sont indiqués sans démonstration dans ma note [P. Assouad, C.R.A.S., Paris, 292 (1981), 921-924]).

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Assouad, Patrick. Densité et dimension. Annales de l'Institut Fourier, Volume 33 (1983) no. 3, pp. 233-282. doi : 10.5802/aif.938. https://aif.centre-mersenne.org/articles/10.5802/aif.938/

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