Let {$X_n$, $n;{in};mathbb{Z}_+^d$} be a field of identically distributed and negatively associated random variables with mean zero and set $S_n;=;{sum}_{k{leq}n};X_k$, $n;{in};mathbb{Z}_+^d$, $d;{geq};2$. We investigate precise asymptotics for ${sum}_n|n|^{r/p-2}P(|S_n|;{geq};{epsilon}|n|^{1/p}$ and ${sum}_n;frac{(log;|n|)^{delta}}{|n|}P(|S_n|;{geq};{epsilon};sqrt{|n|log|n|)}$, ($0;{leq};{delta};{leq};1$) as ${epsilon}{searrow}0$.