Abstract
We prove that a random series ∑ x i γ i , where (x i ) is a sequence of vectors in a Banach space and (γ i ) is a sequence of i.i.d Gaussian random variables is a.s. unconditionally convergent if and only if it is a.s. convergent and the series ∑ x i is unconditionally convergent. A similar statement in which (γ i ) is replaced by a sequence of independent random variables is proved also. We give some applications to Karhunen—Loève representations of Gaussian processes.Key words and Phrasesunconditional convergencerandom seriesBanach spaces
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