ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
Vol. 70, 2 (2001)
pp. 167-175
ON ALMOST SURE CONVERGENCE WITHOUT THE RADON-NIKODYM PROPERTY
N. Bouzar
Abstract.
In this paper we obtain almost sure convergence theorems for
vector-valued uniform amarts and $C$-sequences without assuming
the Radon-Nikodym Property. Specifically, it is shown that if a
limit exists in a weak sense for these martingale generalizations,
then a.s. scalar and strong convergence follow. These results lead
to some new versions of the Ito-Nisio theorem. Convergence results
for random sequences taking values in a weakly compact space are
also presented.
AMS subject classification.
Primary 60G48, 60G40
Keywords.
Vector-valued random variable, stopping time, uniform amart, $C$-sequence,
scalar convergence
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