Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces
Volume 255 / 2020
Abstract
Let $X$ be a symmetric quasi-Banach function space with the Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq 0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$ \Bigl \|\, \Bigl \|\sum _{k=0}^nf_ke_k\Bigr \|_{E}\, \Bigr \|_X $$ admits an equivalent characterization in terms of disjoint copies of $(f_k)_{k=0}^n$ for every $n\ge 0$; in particular, we obtain a deterministic description of $$ \Big \|\, \Big \|\sum _{k=0}^nf_ke_k\Big \|_{\ell _q}\, \Big \|_{L_p} $$ for all $0 \lt p,q \lt \infty ,$ which is the ultimate form of Rosenthal’s inequality. We also consider the case of a $\Delta $-normed symmetric function space $X$, defined via an Orlicz function $\Phi $ satisfying the $\Delta _2$-condition. That is, we provide a formula for “$E$-valued $\Phi $-moments” $\mathbb {E}(\Phi (\|(f_k)_{k\geq 0} \|_E))$, in terms of the sum of disjoint copies of $f_k, k\geq 0.$
