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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.

MSC:
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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