Some deviation inequalities for sums of negatively associated random variables

Let $\{X_i,i\ge 1\}$ be a sequence of negatively associated random variables, and let $\{X_i^{\ast },i\ge 1\}$ be a sequence of independent random variables such that $X_i^{\ast }$ and $X_i$ have the same distribution for each $i$. Denote by $S_k={\sum }_{i=1}^k{X}_i$ and $S_k^{\ast }={\sum }_{i=1}^k{X}_i^{\ast }$ for $k\ge 1$. The well-known results of Shao \cite{Shao2000} sates that $$\mathbb{E}$f(S_n)\leq $\mathbb{E}$f(S_n^\ast)$ for any nondecreasing convex function. Using this very strong property, we obtain a large variety of deviation inequalities for $S_n$