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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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