Johnson-Schechtman and Khinchine inequalities in noncommutative probability theory

Johnson-Schechtman inequalities for noncommutative martingales

In this paper we study Johnson-Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space $E(\mathcal{M})$ of a finite von Neumann algebra $\mathcal{M}$ without making any assumption on the Boyd indices of $E$. We show that we can obtain Johnson-Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random vari…

$Φ$-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 $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let $Φ$ be an Orlicz function with $Δ_2$-condition. We provide an equivalent expression for the quantity $\mathbb{E}(Φ(\sum_{k=1}^n f_k))$ 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 $τ(Φ(\sup^+_{1\leq k\leq n}x_…

Probability inequalities and tail estimates for metric semigroups

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathscr{G}$ a Banach space.) Using our prior work [Ann. Prob. 2017] that extends the Hoffmann-Jorgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the "correct" universal constants in…

强左不变度量半群/幺半群（包括所有李群）的概率不等式

Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups

本研究将Hoffmann-Jorgensen不等式等多个随机不等式及Levy等价性的适用范围从带双不变度量的半群拓展至带强左/右不变度量的半群，覆盖所有李群，并证明了该半群框架与对应幺半群框架等价。