Estimation of a quadratic function and the $p$-Banach–Saks property

Abstract:

Let $E$ be a rearrangement-invariant Banach function space on $[0,1]$, and let $\Gamma (E)$ denote the set of all $p\ge 1$ such that any sequence $\lbrace x_n \rbrace$ in $E$ converging weakly to $0$ has a subsequence $\lbrace y_n \rbrace$ with $\sup _m m^{-1/p}\|\sum _{1\le k\le m} y_n\|<\infty$. The set $\Gamma _i(E)$ is defined similarly, but only sequences $\lbrace x_n \rbrace$ of independent random variables are taken into account. It is proved (under the assumption $\Gamma (E)\ne \lbrace 1\rbrace$) that if $\Gamma _i(E)\setminus \Gamma (E)\ne \varnothing$, then $\Gamma _i(E)\setminus \Gamma (E)=\lbrace 2\rbrace$.