Let {x_n, n\geq 1} be a sequence of i.i.d. random variables and |X_n^(1)|\geq |X_n^(2)|\geq ... \geq |X_n^(n)| be order statistics of |X_1|,|X_2|,...,|X_n|. Set ^{r}S_n=\sum_{i=r+1}^{(1)}X_n^(i) for 0 ≤r≤n-1. Let b_n→\infty, c_n\in R and r=r_n→0. We give the and sufficient conditions when (^{r}S_n-c_n)/b_n couverges to 0 in probability. As a result, we show that Pruitt's conjecture is true.