Let $X_1$, $X_2$, $cdots$ be identically distributed negatively associated random variables with $EX_1;=;0$ and $E|X_1|^3$ < $infty$. In this paper we prove $lim_{{epsilondownarrow}0};frac{1}{-log;epsilon}sumlimits_{n=1}^inftyfrac{1}{n^2}ES_n^2I{|S_n|;{geq};{sigmaepsilon}n};=;2$ and $lim_{epsilondownarrow0};epsilon^{2-p}sumlimits_{n=1}^inftyfrac{1}{n^p}$ $E|S_n|^pI{|S_n|;{geq};{sigmaepsilon}n};=;frac{2}{2-p}$ for 0 < p < 2, where $S_n;=;sumlimits_{i=1}^{n}X_i$ and 0 < $sigma^2;=;EX_1^2;+;sumlimits_{i=2}^{infty}Cov(X_1,;X_i)$ < $infty$. We consider some results of i.i.d. random variables obtained by Liu and Lin(2006) under negative association assumption.