Limit infimum results for subsequences of partial sums and random sums

Let {Xn,n⩾1} be a sequence of i.i.d. valued random variables with distribution function F. Let Sn=∑k=1nXk,n⩾1. When F belongs to the domain of normal attraction of a Stable law, the law of iterated logarithm has been obtained for subsequences of (Sn) and extended to random subsequences.