Geodesic
On the convergence of certain sums of independent random elements
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 77-81.

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In this note we investigate the relationship between the convergence of the sequence $\{S_{n}\}$ of sums of independent random elements of the form $S_{n}=\sum_{i=1}^{n}\varepsilon_{i}x_{i}$ (where $\varepsilon_{i}$ takes the values $\pm\,1$ with the same probability and $x_{i}$ belongs to a real Banach space $X$ for each $i\in \Bbb N$) and the existence of certain weakly unconditionally Cauchy subseries of $\sum_{n=1}^{\infty}x_{n}$.
Classification : 46B09, 46B15, 60B12
Keywords: independent random elements; copy of $c_{0}$; Pettis integrable function; perfect measure space 
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Ferrando, J. C. On the convergence of certain sums of independent random elements. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 1, pp. 77-81. http://geodesic.mathdoc.fr/item/CMUC_2002__43_1_a6/