Φ-moment inequalities for independent and freely independent random variables

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• 作者：Yong Jiao^a ; ^{jiaoyong@csu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Fedor Sukochev^b ; ^{f.sukochev@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Guangheng Xie^a ; ^{xiegh@csu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Dmitriy Zanin^b ; ^{d.zanin@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46L52 ; 46L53 ; 47A30 ; secondary ; 60G42
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：270
• 期：12
• 页码：4558-4596
• 全文大小：605 K

文摘

This paper is devoted to the study of Φ-moments of sums of independent/freely independent random variables. More precisely, let ${(f_k)}_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let Φ be an Orlicz function with Δ₂${\mathrm{\Delta }}_2$-condition. We provide an equivalent expression for the quantity $\mathbb{E}(\mathrm{\Phi }({\sum }_{k=1}^n{f}_k))$ in term of the sum of disjoint copies of the sequence ${(f_k)}_{k=1}^n$. We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of $\tau (\mathrm{\Phi }({\sup }_{1\le k\le n}^{+}{x}_k))$ for positive freely independent random variables and also present some new results on free Johnson–Schechtman inequalities in the quasi-Banach symmetric operator space.