Abstract
This paper studies series of independent random variables in rearrangement invariant spaces X on [0, 1]. Principal results of the paper concern such series in Orlicz spaces exp(Lp), 1 ~ p ~ c~ and Lorentz spaces A¢. One by-product of our methods is a new (and simpler) proof of a result due to W. B. Johnson and G. Schechtman that the assumption Lp C X, p < oc is sufficient to guarantee that convergence of such series in X (under the side condition that the sum of the measures of the supports of all individual terms does not exceed 1) is equivalent to convergence in X of the series of disjoint copies of individual terms. Furthermore, we prove the converse (in a certain sense) to that result.
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