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An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales

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An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales SÉMINAIRE DE PROBABILITÉS (STRASBOURG) MAURIZIO PRATELLI An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales Séminaire de probabilités (Strasbourg), tome 33 (1999), p. 334-338. <http://www.numdam.org/item?id=SPS_1999__33__334_0> © Springer-Verlag, Berlin Heidelberg New York, 1999, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ An alternative proof of a theorem of Aldous concerning convergence in distribution for martingales. Maurizio Pratelli We consider regular right continuous stochastic processes X = de- fined on the finite time interval [0,1]: let P~ be the distribution of X on the canonical Skorokhod space D = ® ( (0,1~; R) of "cadlag" paths. We consider on D , besides the usual Skorokhod topology referred as S-topology (Jacod-Shiryaev is perhaps the best reference for our purposes, see [4]), the "pseudo- path" or MZ-topology: we refer to the paper of Meyer-Zheng ([6]) for a complete account of this rather neglected topology (see also Kurtz [5]). We will use the notation Xn ~S X (respectively X’~ X ) to indicate that the probabilities converge strictly to P~ when the space D is endowed respectively with the S- or the MZ-topology. We will write also X’~ to indicate that all finite dimensional distributions of converge to those of ~ The following theorem holds true: Theorem. Let be a sequence of martingales, and M a conti

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