We consider independent and identically distributed random variables X,X_n,n≥1 with density f_X(x)=c(log x)^λx^-2I(x>1), where λ>-1. Hence EX=∞. We examine which type of random weights a_n,n≥1 that will allow us to establish our generalized one-sided laws of the iterated logarithm for sums of the form