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 $1\le r\le 2$ 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 $a_j\le x_j\le b_j$ for some real numbers $a_j<b_j$ and ${\mathcal{E}}_{\mathfrak{N}}(x_j)=\mu$ for some $\mu \ge 0$ and all $1\leq j\leq n$\mathrm{.}Thenforany$t>o$ 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*}