ON ALMOST SURE CONVERGENCE OF CLASSES OF MULTIVALUED ASYMPTOTIC MARTINGALES.

摘要

Let ' be the dual of a Banach space . We assume that ' is separable. Let K denote the class of all non-empty, convex, weak*-compact sets in '. If (DELTA) is the Hausdorff's metric on K, then (K, (DELTA)) is a complete metric space. However, K need not be separable in general. A multivalued random variable defined on a probability space ((OMEGA), F, P) is a measurable map from (OMEGA) to K where K is equipped with the Borel (sigma)-field given by the metric (DELTA). Using the one-one correspondence between the elements of K and the continuous sub-linear functionals on , one defines the expectation and the conditional expectation of a multivalued random variable. This has been discussed in a paper by Neveu who also defined multivalued martingales and proved the almost sure convergence of them under proper boundedness conditions.;Multivalued amarts of infinite order are characterized in terms of convergence in Pettis distance and also in terms of Riesz approximation by martingales the Pettis distance of which from the original process goes to zero. This concept was first introduced by Luu. Even real-valued amarts of infinite order need not converge almost surely. We extend some results of Edgar and Sucheston about real-valued amarts to multivalued amarts. A theorem of Bellow about strong convergence of amarts and dimensionality of Banach spaces is extended to multivalued amarts by Assani.;Finally we prove almost sure convergence of real-valued L log('m-1)L-bounded martingales, (X(,t))(,t) (,m) where (X(,t)) is also a k-martingale for every k (LESSTHEQ) m - 1. This extends a result of Millet and Sucheston to more than two parameters. Using this we prove a similar theorem about multivalued multiparameter martingales.;The purpose of this paper is to study wider classes of multivalued processes and investigate their almost everywhere convergence. Using results of Chacon - Sucheston and Millet - Sucheston we prove almost sure weak*-convergence of multivalued weak*-amarts and weak*-pramarts respectively. Both are extensions of the weak*-convergence of multivalued martingales. We also prove strong convergence of pramarts under the assumption the limiting random variable takes values in a separable subspace of K. This extends a similar convergence theorem of martingales proved under identical assumptions.