Rocky Mountain Journal of Mathematics

Inequalities for sums of random variables in noncommutative probability spaces

Ghadir Sadeghi and Mohammad Sal Moslehian

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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 \[ {\rm Prob}\bigg (\bigg |\sum _{j=1}^n x_j-n\mu \bigg |\geq t\bigg )\leq 2 \exp \bigg \{\frac {-2t^2}{\sum _{j=1}^n(b_j-a_j)^2}\bigg \}. \]

Article information

Rocky Mountain J. Math., Volume 46, Number 1 (2016), 309-323.

First available in Project Euclid: 23 May 2016

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Zentralblatt MATH identifier

Primary: 46L53: Noncommutative probability and statistics 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B11: Probability theory on linear topological spaces [See also 28C20]

Noncommutative probability space von Neumann algebra noncommutative Rosenthal inequality noncommutative Bennett inequality noncommutative Hoeffding inequality


Sadeghi, Ghadir; Moslehian, Mohammad Sal. Inequalities for sums of random variables in noncommutative probability spaces. Rocky Mountain J. Math. 46 (2016), no. 1, 309--323. doi:10.1216/RMJ-2016-46-1-309.

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