# Article

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Keywords:
independent random elements; copy of $c_{0}$; Pettis integrable function; perfect measure space
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 $i\in \Bbb N$) and the existence of certain weakly unconditionally Cauchy subseries of $\sum_{n=1}^{\infty}x_{n}$.
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