Series of independent random variables in rearrangement invariant spaces: an operator approach (Q1775470)

This article studies series of independent random variables in rearrangement invariant spaces on \([0,1]\). Its principal concern is the following question: For which rearrangement invariant spaces \(X\) and \(Y\) on \([0,1]\) does there exist a constant \(C\) such that for every sequence \((f_k)\) of independent mean zero random variables in \(X\) (with an additional mild condition), one has that \[ \| \sum_k f_k\| _Y\leq C \| \sum_k \bar f_k\| _X\,, \] where \((\bar f_k)\) is a disjoint copy of \((f_k)\)? The authors prove that if \(X\) is the Orlicz space \({\exp}(L_p)\) (\(1\leq p\leq\infty\)), then among all \(Y\) satisfying the foregoing inequality there exists a minimal one (which is also an Orlicz space). On the other hand, the authors characterize Lorentz spaces \(\Lambda_\psi[0,1]\) for which the correspondence \(f_k\leftrightarrow \bar f_k\) extends to an isomorphism between the closed linear spans \([f_k]\) and \([\bar f_k]\) in \(\Lambda_\psi[0,1]\). This article is closely related to a previous one by \textit{W. B. Johnson} and \textit{G. Schechtman} [Ann. Probab. 17, No.~2, 789--808 (1989; Zbl 0674.60051)]. In particular, it includes the converse to one of the main results in the latter article; that is, if the correspondence \(f_k\leftrightarrow \bar f_k\) extends to an isomorphism between the closed linear spans \([f_k]\) and \([\bar f_k]\) in \(X\), then \(X\) contains an \(L_p\) (\(p<\infty\)).