Rosenthal's inequalities: $Δ-$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 $(f_k){k\geq0}\subset X$ is a sequence of independent random variables. We present a necessary and sufficient condition on $X$ such that the quantity $$\Big|\ \Big|\sum{k=0}^nf_ke_k\Big|_{E}\ \Big|X$$ 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 $$\Big|\ \Big|\sum {k=0}^nf_ke_k\Big|{\ell_q}\ \Big|{L_p}$$ 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 $\mathbb{E}\big(\Phi\big(\big|(f_k){k\geq0} \big|_E \big)\big)$, in terms of the sum of disjoint copies of $f_k, k\geq0.$