Open Access
2016 Inequalities for sums of random variables in noncommutative probability spaces
Ghadir Sadeghi, Mohammad Sal Moslehian
Rocky Mountain J. Math. 46(1): 309-323 (2016). DOI: 10.1216/RMJ-2016-46-1-309


In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq 2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak {M}, \tau )$ be a noncommutative probability space, $\mathfrak {N}$ be a von Neumann subalgebra of $\mathfrak {M}$ with the corresponding conditional expectation $\mathcal {E}_{\mathfrak {N}}$ and let subalgebras $\mathfrak {N}\subseteq \mathfrak {A}_j\subseteq \mathfrak {M}\,\,(j=1, \cdots , n)$ be successively independent over $\mathfrak {N}$. Let $x_j\in \mathfrak {A}_j$ be self-adjoint such that $a_j\leq x_j\leq b_j$ for some real numbers $a_j\lt b_j$ and $\mathcal {E}_{\mathfrak {N}}(x_j)=\mu $ for some $\mu \geq 0$ and all $1\leq j\leq n$. Then for any $t>o$ it holds that \[ {\rm Prob}\bigg (\bigg |\sum _{j=1}^n x_j-n\mu \bigg |\geq t\bigg )\leq 2 \exp \bigg \{\frac {-2t^2}{\sum _{j=1}^n(b_j-a_j)^2}\bigg \}. \]


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Ghadir Sadeghi. Mohammad Sal Moslehian. "Inequalities for sums of random variables in noncommutative probability spaces." Rocky Mountain J. Math. 46 (1) 309 - 323, 2016.


Published: 2016
First available in Project Euclid: 23 May 2016

zbMATH: 1357.46057
MathSciNet: MR3506091
Digital Object Identifier: 10.1216/RMJ-2016-46-1-309

Primary: 46L53 , 47A30
Secondary: 60B11 , 60B20

Keywords: noncommutative Bennett inequality , noncommutative Hoeffding inequality , Noncommutative probability space , noncommutative Rosenthal inequality , von Neumann algebra

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.46 • No. 1 • 2016
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