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

Let {Xi,i≥1}{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=∑i=1nXi,Vn2=∑i=1nXi2. Then an almost sure central limit theorem for self-normalized partial sums Sn/VnSn/Vn is studied under a mild condition in this paper.