Decompositions and limits for martingale-like sequences in Banach spaces
The notion of games which become fairer with time (see definition in the next section) was first introduced by \textit{L. H. Blake} [Pac. J. Math. 35, 279-283 (1970; Zbl 0218.60050)], \textit{J. Neveu} [Martingales à temps discrets. (1972; see the review 1975 English edition, Zbl 0345.60026)] proved that every \(L^ 1\)-bounded Banach space-valued martingale convergs (strongly) to zero a.e. if it converges scalarly to zero a.e. In this note we extend also this well-known result of Neveu to Banach space-valued games which become fairer with time. Finally, using this extension and the structure results of \textit{M. Talagrand} [Ann. Probab. 13, 1192-1203 (1985; Zbl 0582.60055)], we show that every \(L^ 1\)-bounded Banach space-valued games fairer with time \((X_ n)\) can be written in a unique form: \(X_ n=M_ n+P_ n\), where \((M_ n)\) is a uniformly integrable martingale and \((P_ n)\) is a game fairer with time that goes to zero in probability. Hence, every such a game fairer with time taking values in a Banach space having the Radon- Nikodym property, converges in probability. Thus, the main convergence problem for games which become fairer with time is solved.
scientific article; zbMATH DE number 4129716