Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable law

Let (Xn) be a sequence of independent and identically distributed non-negative valued random variables defined over a common probability space and let F denote the common distribution function. Set Sn=X1+X2+⋯+Xn and denote inf{x;n(1−F(x))≤1} by Bn,n≥1. Assuming that F belongs to the domain of partial attraction of a positive semi-stable law, we obtain the set of almost sure limit functions of Bn−1S[nt]1loglogn,t∈[0,1], under M1-topology.