Khinchin inequality and Banach–Saks type properties in rearrangement-invariant spaces
Tom 191 / 2009
Studia Mathematica 191 (2009), 101-122
MSC: 46E30, 46B20.
DOI: 10.4064/sm191-2-1
Streszczenie
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}\leq Cn^{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.
