Abstract: We try to answer a question about the increments of sums of i.i.d,random variables,raised by Hanson and Russo,as condition（5.3b）:α_n/logn→∞ in their paper.Can this condition be relaxed if more is assumed about the tails of the distributions of the random variables?We also consider relaxation of their condition（5.7）:（logα_n）/n→0,The results are similar to the well-known Erdös-Rényi’s new law of large numbers.