Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables

Let {X,Xn,n≥1}{X,Xn,n≥1} be a sequence of i.i.d. random variables and set Sn=∑i=1nXi. NN is the standard normal random variable, then for d>0d>0 and β>0β>0, we show that limε↘0ε2β/d∑n=3∞(loglogn)β−1n3/2lognE{|Sn|−εσn(loglogn)d/2}+=dσE|N|2β/d+1β(2β+d) holds if and only if EX=0,EX2=σ2.