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On further classes of martingale-like sequences and some decomposition and convergence theorems

Published online by Cambridge University Press:  01 October 1999

Dinh Quang Luu
Hanoi Institute of Mathematics, P.O. Box 631 Bo-Ho, Hanoi, Vietnam Actual address: IM - Stefan Banach Center, ul. Mokotowska 25, skr. 137, 00-950 Warsaw, Poland
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It is known that the class of mils generalizes that of pramarts and martingales in the limit. Also every Banach space-valued mil (Xn) with lim infnE(‖Xn‖)<∞ can be written in a unique form: $X_n=M_n+P_n(n\in\rm{N})$, where $(M_n)$ is a uniformly integrable martingale and $(P_n)$ converges to zero a.s. in norm. We shall show that this result still holds for a class which essentially generalizes that of mils. Another class of Banach space-valued martingale-like sequences, still containing all pramarts is defined and shown to have the decomposition above under the following much weaker condition: $\rm{lim inf}_{r\inT}E(\VertX_{\tau}\Vert)<\infty$, where T denotes the set of all bounded stopping times.

1991 Mathematics Subject Classification. 60G48, 60B11.

Research Article
1999 Glasgow Mathematical Journal Trust