Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces

Abstract

{\it We study the class of all rearrangement-invariant (=r.i.) function spaces $E$ on $[0,1]$ such that there exists $0<q<1$ for which $\Vert {\sum }_{{}_{k=1}}^n{\xi }_k{\Vert }_E\le C{n}^q$, where $\{{\xi }_k{\}}_{k\ge 1}\subset E$ is an arbitrary sequence of independent identically distributed symmetric random variables on $[0,1]$ and $C>0$ does not depend on $n$. We completely characterize all Lorentz spaces having this property and complement classical results of Rodin and Semenov for Orlicz spaces $exp(L_p)$, $p\ge 1$. We further apply our results to the study of Banach-Saks index sets in r.i. spaces.