Φ-moment inequalities for independent and freely independent random variables

This paper is devoted to the study of Φ-moments of sums of independent/freely independent random variables. More precisely, let (fk)k=1n be a sequence of positive (symmetrically distributed) independent random variables and let Φ be an Orlicz function with Δ2Δ2-condition. We provide an equivalent expression for the quantity E(Φ(∑k=1nfk)) in term of the sum of disjoint copies of the sequence (fk)k=1n. We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of τ(Φ(sup1≤k≤n+⁡xk)) for positive freely independent random variables and also present some new results on free Johnson–Schechtman inequalities in the quasi-Banach symmetric operator space.