Rosalsky, Andrew; Le Van Thanh; Volodin, Andrei I. On complete convergence in mean of normed sums of independent random elements in Banach spaces. (English) Zbl 1087.60009 Stochastic Anal. Appl. 24, No. 1, 23-35 (2006). Let \((T_n)\) be a sequence of random elements in a separable Banach space. For \(p>0\), \(T_n\) is said to converge completely to \(0\) in mean of order \(p\) if \(\sum_1^{\infty}E\| T_n\| ^p<\infty\). This mode of convergence was introduced by Y.S. Chow [Bull. Inst. Math., Acad. Sin. 16, No. 3, 177–201 (1988; Zbl 0655.60028)] for real-valued random variables. In the paper under review conditions are provided under which a normed sum of independent random elements in a Rademacher type \(p\) Banach space converges completely to \(0\) in mean of order \(p\). An example, showing that the conditions are sharp, is given. Moreover, these conditions provide an exact characterization of Rademacher type \(p\) Banach spaces. Illustrative examples are presented. Reviewer: Anatolij M. Plichko (Krakow) Cited in 1 ReviewCited in 13 Documents MSC: 60B11 Probability theory on linear topological spaces 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F15 Strong limit theorems 60F25 \(L^p\)-limit theorems Keywords:complete convergence in mean; normed sums of independent random elements; Rademacher type \(p\) Banach spaces Citations:Zbl 0655.60028 PDFBibTeX XMLCite \textit{A. Rosalsky} et al., Stochastic Anal. Appl. 24, No. 1, 23--35 (2006; Zbl 1087.60009) Full Text: DOI References: [1] DOI: 10.1080/07362998708809104 · Zbl 0617.60028 · doi:10.1080/07362998708809104 [2] Chow Y.S., Bull. Inst. Math. Acad. Sinica 16 pp 177– (1988) [3] Chow Y.S., Probability Theory: Independence, Interchangeability, Martingales, 3. ed. (1997) [4] Etemadi N., Sankhyā Ser. A 47 pp 209– (1985) [5] DOI: 10.1214/aop/1176996029 · Zbl 0368.60022 · doi:10.1214/aop/1176996029 [6] DOI: 10.1073/pnas.33.2.25 · Zbl 0030.20101 · doi:10.1073/pnas.33.2.25 [7] DOI: 10.1214/aop/1176995149 · Zbl 0399.60007 · doi:10.1214/aop/1176995149 [8] Scalora F.S., Pacific J. Math. 11 pp 347– (1961) · Zbl 0114.07702 · doi:10.2140/pjm.1961.11.347 [9] DOI: 10.1090/S0002-9947-1956-0079851-X · doi:10.1090/S0002-9947-1956-0079851-X [10] Taylor R.L., Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces (1978) · Zbl 0443.60004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.