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

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