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

TL;DR

This paper characterizes Lorentz spaces with a specific inequality involving sums of i.i.d. symmetric variables, extending classical results and applying findings to Banach-Saks index sets in rearrangement-invariant spaces.

Contribution

It provides a complete characterization of Lorentz spaces satisfying a Khinchin-type inequality and extends classical results to new classes of rearrangement-invariant spaces.

Findings

Lorentz spaces with the inequality are fully characterized.

Classical results for Orlicz spaces are extended.

Applications to Banach-Saks index sets are demonstrated.

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}\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.