Let {$X_k;k{in}N^d$} be centered and identically distributed random field which is asymptotically negative dependent in a certain case. In this note we prove that for $p{alpha}$ > 1 and ${alpha}$ > ${frac{1}{2}}$ $E{mid}X_1{mid}^p(log^+{mid}X_1{mid}^{d-1})$ < ${infty}$ if and only if ${sum}_n{mid}n{mid}^{p{alpha}-2}P$($max_{1{leq}k{leq}n{mid}S_k{mid}}$ > ${epsilon}{mid}n{mid}$) < ${infty}$ for all ${epsilon}$ > 0, where log$^+$x = max{1,log x}.