Let {X, $X_{i};;i{geq}1$} be a sequence of independent and identically distributed positive random variables. Denote $S_n= sumarray\_{i=1}^nX_i$ and $Sarray\_n^{(k)}=S_n-X_k$ for n ${geq}$1, $1{leq}k{leq}n$. Under the assumption of the finiteness of the second moments, we derive a type of the law of the iterated logarithm for $Sarray\_n^{(k)}$ and the limit point set for its certain normalization.