Khinchin inequality and Banach-Saks type properties in rearrangement-invariant spaces
{\it We study the class of all rearrangement-invariant (=r.i.) function spaces $E$ on $[0,1]$ such that there exists $0 0$ 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.