Let ${X_i{mid}i{geq}1}$ be a strictly stationary sequence of associated random variables with mean zero and let ${sigma}^2=EX_1^2+2sumlimits_{j=2}^infty{EX_1}{X_j}$ with 0 < ${sigma}^2$ < ${infty}$. Set $S_n={sumlimits^n_{i=1}^{X_i}$, the precise asymptotics for ${varepsilon}^{{frac{2(r-p)}{2-p}}-1}sumlimits_{n{geq}1}n^{{frac{r}{p}}-{frac{1}{p}}+{frac{1}{2}}}P({mid}S_n{mid}{geq}{varepsilon}n^{{frac{1}{p}}})$,${varepsilon}^2sumlimits_{n{geq}3}{frac{1}{nlogn}}p({mid}Sn{mid}{geq}{varepsilonsqrt{nloglogn}})$ and ${varepsilon}^{2{delta}+2}sumlimits_{n{geq}1}{frac{(loglogn)^{delta}}{nlogn}}p({mid}S_n{mid}{geq}{varepsilonsqrt{nloglogn}})$ as ${varepsilon}{searrow}0$ are established under the suitable conditions.