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## Studia Mathematica

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## Rosenthal’s inequalities: ${\Delta }$-norms and quasi-Banach symmetric sequence spaces

### Volume 255 / 2020

Studia Mathematica 255 (2020), 55-81 MSC: Primary 46E30; Secondary 60G50. DOI: 10.4064/sm190503-28-10 Published online: 27 May 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 $f_k, k\geq 0.$

#### Authors

• Yong JiaoSchool of Mathematics and Statistics
Central South University
Changsha 410075, People’s Republic of China
e-mail
• Fedor SukochevSchool of Mathematics and Statistics
University of NSW
Sydney, 2052, Australia
e-mail
• Guangheng XieSchool of Mathematics and Statistics
Central South University
Changsha 410075, People’s Republic of China
e-mail
• Dmitriy ZaninSchool of Mathematics and Statistics
University of NSW
Sydney, 2052, Australia
e-mail

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