Rosenthal's inequalities: $\Delta -$norms and quasi-Banach symmetric sequence spaces

Abstract

Let $X$ be a symmetric quasi-Banach function space with Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k{)}_{k\ge 0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$\Vert \Vert \sum _{k=0}^n{f}_k{e}_k{\Vert }_E{\Vert }_X$$ admits an equivalent characterization in terms of disjoint copies of $(f_k{)}_{k=0}^n$ for every $n\ge 0$; in particular, we obtain the deterministic description of $$\Vert \Vert \sum _{k=0}^n{f}_k{e}_k{\Vert }_{{\ell }_q}{\Vert }_{L_p}$$ for all $0<p,q<\infty ,$ 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 \lq\lq $E$-valued $\Phi$-moments\rq\rq, namely the quantity $\mathbb{E}\left(\Phi \right(\Vert (f_k{)}_{k\ge 0}{\Vert }_E))$, in terms of the sum of disjoint copies of $f_k,k\ge 0.$