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Unconditional convergence of random series and the geometry of Banach spaces. (English) Zbl 0965.60015
Let \(X\) be a real Banach space and \((\Omega,{\mathcal A},P)\) a probability space. Let \(\{\xi_{k}\}\) be a sequence of random elements in \(X\). A random series \(\sum_{k=1}^{\infty}\xi_{k}\) is said to be a.s. unconditionally convergent in \(X\) if there exists a set \(\Omega_{0}\in{\mathcal A}\) with probability one such that the series \(\sum_{k=1}^{\infty}\xi_{k}(\omega)\) converges unconditionally in \(X\) for any \(\omega\in\Omega_{0}\). The author gives a sufficient condition on a sequence \(\{\zeta_{k}\}\) of real random variables, under which the a.s. unconditional convergence of the series \(\sum_{k=1}^{\infty}a_{k}\zeta_{k}\), \(a_{k}\in X\), implies the unconditional convergence of the series \(\sum_{k=1}^{\infty}a_{k}\) in \(X\). Some connections of the a.s. unconditional convergence with the geometry of Banach spaces are established.

60B11 Probability theory on linear topological spaces
60B05 Probability measures on topological spaces
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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