Precise rates in the generalized law of the iterated logarithm

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with EX=0 and 00 and an=o((loglogn)−d), we show the precise rates in the generalized law of the iterated logarithm for a kind of weighted infinite series of P{|Sn|≥(ε+an)σn(loglogn)d}.