Law of iterated logarithm and almost sure limit points for random sums

Let {Xn,n≥1} be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi-Stable law with index α,0<α<2, we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189-194], where they obtained Chover's form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.