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

Let {X,Xn;n⩾1} be a sequence of i.i.d. random variables with EX=0EX=0 and EX2=σ2<∞EX2=σ2<∞. Set Sn=∑k=1nXk, Mn=maxk⩽n|Sk|Mn=maxk⩽n|Sk|, n⩾1n⩾1. Let r>1r>1, then we obtainlimε↘r−11−log(ε2−(r−1))∑n=1∞nr−2−1/2E{Mn−σε2nlogn}+=2σ(r−1)2π holds, if and only if EX=0EX=0, EX2=σ2<∞EX2=σ2<∞ and E(|X|2r/r(log|X|))<∞E(|X|2r/(log|X|)r)<∞.