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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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