On convergence of vector-valued weak amarts and pramarts (Q863776)

A sequence \((X_n)\) of random elements in a Banach space \(B\) is called essentially (essentially weakly) tight if and only if for every \(\varepsilon>0\) there exists a (weakly) compact subset \(K\) of \(B\) such that \[ P\left(\bigcap^\infty_{n=1}\{X_n\in K\}\right)>1-\varepsilon. \] Note that essential tightness. In fact, \textit{G. Krupa} and \textit{W. Ziȩba} [Commentat. Math. Univ. Carol. 37, No. 3, 641--650 (1996; Zbl 0881.60003)] proved that an \(L^1\)-bounded strong amart in a Banach space converges almost surely if and only if it is essentially tight. Based on methods due to \textit{W. J. Davis, N. Ghoussoub, W. B. Johnson, S. Kwapien} and \textit{B. Maurey} [in: Probability in Banach spaces, 6. Prog. Probab. Stat. 20, 41--50 (1990; Zbl 0709.60007)] and \textit{F. Bouzar} [Acta Math. Univ. Comen., New Ser. 70, No. 2, 167--175 (2001; Zbl 1007.60032)], the main convergence results of these authors for amarts are extended to weak amarts and pramarts of Pettis integrable functions in Banach spaces without the Radon-Nikodým property.