Φ-moment inequalities for independent and freely independent random variables

2016/08/26 by Yong Jiao, Jiao, Yong, Fedor Sukochev +5
Mathematics · #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Probability (math.PR) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1608.07368

openalex created_date 2016/06/24 · openalex publication_date 2016/08/26 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the study of Φ-moments of sums of independent/freely independent random variables. More precisely, let (f_k)_k=1ⁿ be a sequence of positive (symmetrically distributed) independent random variables and let Φ be an Orlicz function with Δ₂-condition. We provide an equivalent expression for the quantity 𝔼(Φ(∑_k=1ⁿ f_k)) in term of the sum of disjoint copies of the sequence (f_k)_k=1ⁿ. We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of τ(Φ(sup⁺_{1≤ k≤ 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.

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