Let {$X,;X_n;n{geq}1$} be a sequence of i.i.d. random variables. Set $S_n=X_1+X_2+{cdots}+X_n,;M_n=max_{k{leq}n}|S_k|,;n{geq}1$. Then we obtain that for any -1<b<1/2, $limlimits_{{varepsilon}{searrow}0};{varepsilon}^{2b+2}sumlimits_{n=1}^infty;{frac {(log;n)^b}{n^{3/2}};E{M_n-{varepsilon}{sigma}sqrt{n;log;n}+=frac{2sigma}{(b+1)(2b+3)};E|N|^{2b+3}sumlimits_{k=0}^infty;{frac{(-1)^k}{(2k+1)^{2b+3}$ if and only if EX=0 and $EX^2={sigma}^2<{infty}$.