Processus linéaires, processus généralisés
Annales de l'Institut Fourier, Tome 17 (1967) no. 1, pp. 1-92.

Dans le premier chapitre, on présente certains espaces topologiques (espaces standards) généralisant les espaces polonais et sur lesquels on construit un calcul des probabilités. La plupart des espaces fonctionnels usuels sont des espaces standards. On étudie les convergences des mesures et des variables aléatoires liées à de tels espaces.

Dans le deuxième chapitre, on présente les processus linéaires : famille de variables aléatoires numériques indexée par les éléments d’un espace vectoriel vérifiant certaines conditions linéaires. On étudie les régularités des processus linéaires.

Le chapitre 3 applique les résultats précédents à l’espace des distributions. On montre en particulier que la fonctionnelle caractéristique est un outil suffisant pour l’étude des distributions aléatoires et de leurs convergences ; on généralise en particulier le théorème de P. Lévy sur la convergence en loi des variables aléatoires usuelles.

Le chapitre 4 est un chapitre d’exemples.

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Fernique, Xavier. Processus linéaires, processus généralisés. Annales de l'Institut Fourier, Tome 17 (1967) no. 1, pp. 1-92. doi : 10.5802/aif.249. https://www.numdam.org/articles/10.5802/aif.249/

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[1a] N. Bourbaki, op. cit., 2e édition, Hermann, Paris (1951).

[2] N. Bourbaki, op. cit., Livre 3, Ch. 9, 2e édition (1958).

[3] N. Bourbaki, op. cit., Livre 5, Ch. 1.2, 2e édition (1952).

[4] N. Bourbaki, op. cit., Livre 5, Ch. 3.4, 2e édition (1955).

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[13] I. M. Gelfand, Dokl. Akad. Nauk. USSR, 100 (1955), 853. | Zbl

[14] M. Loeve, Probability Theory, Van Nostrand, New-York (1960). | MR | Zbl

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