# Inequalities for sums of random variables in noncommutative probability
spaces

Research paper by **Ghadir Sadeghi, Mohammad Sal Moslehian**

Indexed on: **22 Jun '14**Published on: **22 Jun '14**Published in: **Mathematics - Operator Algebras**

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#### Abstract

In this paper, we establish an extension of a noncommutative Bennett
inequality with a parameter $1\leq r\leq2$ 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 $a_j<b_j$
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 \begin{eqnarray*} {\rm
Prob}\left(\left|\sum_{j=1}^n x_j-n\mu\right|\geq t\right)\leq 2
\exp\left\{\frac{-2t^2}{\sum_{j=1}^n(b_j-a_j)^2}\right\}. \end{eqnarray*}