{"paper":{"title":"$\\Phi$-moment inequalities for independent and freely independent random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dmitriy Zanin, Fedor Sukochev, Guangheng Xie, Yong Jiao","submitted_at":"2016-08-26T06:04:12Z","abstract_excerpt":"This paper is devoted to the study of $\\Phi$-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 $\\Phi$ be an Orlicz function with $\\Delta_2$-condition. We provide an equivalent expression for the quantity $\\mathbb{E}(\\Phi(\\sum_{k=1}^n f_k))$ 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(\\Phi(\\sup"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07368","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}