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Some new Chacon-Edgar-type inequalities for stochastic processes, and characterizations of Vitali-conditions

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  • Law
  • Mathematics

Abstract

Some new Chacon-Edgar-type inequalities for stochastic processes, and characterizations of Vitali-conditions ANNALES DE L’I. H. P., SECTION B L. EGGHE Some new Chacon-Edgar-type inequalities for stochastic processes, and characterizations of Vitali-conditions Annales de l’I. H. P., section B, tome 16, no 4 (1980), p. 327-337. <http://www.numdam.org/item?id=AIHPB_1980__16_4_327_0> © Gauthier-Villars, 1980, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb), implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/legal.php). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Some new Chacon-Edgar-type inequalities for stochastic processes, and characterizations of Vitali-conditions L. EGGHE Limburgs Universitair Centrum, Universitaire Campus, B-3610 Diepenbeek (Belgium) Ann. Inst. Henri Poincaré, Vol. XVI, n° 4, 1980, p. 327-337. Section B : Calcul des Probabilités et Statistique. ABSTRACT. In this paper we prove that Edgar’s main inequality in [5] extends to stochastic processes (Xi, where satisfies the Vitali- condition V, when we use the notion essential lim sup in this inequality. We also prove that the inequality is right without V, but then using the notion stochastic lim sup. At the same time, we also generalise some maxi- mal inequalities, proved in [12] and some convergence results in [10]. § 1. INTRODUCTION, TERMINOLOGY AND NOTATION The main result in [5] can be stated as follows : THEOREM 1.la [5]. - Let (Q, F, P) be a probability space and E a sepa- rable dual Banach space. Let Fn) be an adapted Ll-bounded sequence of E-valued Bochner integrable random variables. Then, if T denotes t

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