Inequalities for sums of random variables in noncommutative probability spaces

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 $(\mathfrak{M}, τ)$ 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 $\mathcal{E}_{\mathfrak{N}}(x_j)=μ$ for some $μ\geq 0$ 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-nμ\right|\geq t\right)\leq 2 \exp\left\{\frac{-2t^2}{\sum_{j=1}^n(b_j-a_j)^2}\right\}. \end{eqnarray*}

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