## On the convergence of certain sums of independent random elements.(English)Zbl 1090.46009

Summary: 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 $$i\in \mathbb N$$) and the existence of certain weakly unconditionally Cauchy subseries of $$\sum _{n=1}^{\infty }x_{n}$$.

### MSC:

 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 46B09 Probabilistic methods in Banach space theory
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