Almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces
This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. Sufficient conditions for almost sure convergence and complete convergence in the sense defined by \textit{P. L. Hsu} and \textit{H. Robbins} [Proc. Natl. Acad. Sci. USA 33, 25--31 (1947; Zbl 0030.20101)] and \textit{P. Erdős} [Ann. Math. Stat. 20, 286-291 (1949; Zbl 0033.29001)] are provided. Examples showing that the conditions cannot he removed or weakened are given. It is also demonstrated that some of the known theorems in the literature [\textit{D. Li}, \textit{M. B. Rao}, \textit{T. Jiang} and \textit{X. Wang}, J. Theor. Probab. 8, No. 1, 49--76 (1995; Zbl 0814.60026); \textit{X. Wang}, \textit{M. B. Rao} and \textit{X. Yang}, Stochastic Anal. Appl. 11, No. 1, 115--132 (1993; Zbl 0764.60037)] are special cases of the result, published in this paper.
scientific article; zbMATH DE number 5984261