Precise rates in the law of logarithm for i.i.d. random variables

Let {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables. Set Sn = X1 + X2 + … + Xn and Mn = maxk≤n |Sk|, n ≥ 1. By using the strong approximation method, we obtain that for any −1 < b ≤ 1, lim⁡ε↘0ε2b+2∑n=1∞(log⁡n)bnP(Mn≥εσnlog⁡n)=2E|N|(2b+2)b+1∑k=0∞(−1)k(2k+1)2b+2 if and only if Ex = 0 and Ex2 < ∞, which strengthen and extend the result of Gut and Spǎtaru [1], where N is the standard normal random variable. Furthermore, L2 convergence and a.s. convergence are also discussed.