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

Abstract

It is known that the class of mils generalizes that of pramarts and martingales in the limit. Also every Banach space-valued mil (X_n) with lim inf_nE(‖X_n‖)<∞ 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.