An almost sure central limit theorem for self-normalized partial sums

Let {Xi, i ≥ 1} be a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote Sn = Σni =1 Xi, V2 n = Σn i=1 X2 i . Then an almost sure central limit theorem for self-normalized partial sums Sn/Vn is studied under a mild condition in this paper.