Let $\xi_1,\xi_2,\dots,\xi_n$ be independent random variables, $$Q_k\{l\}=\mathop{\sup}\limits_x\mathbf P\{x\leq\xi _k\leq x+l\},\\Q(L)=\mathop{\sup}\limits_x\mathbf P\{{x\leq\xi_1+\cdots+\xi_n\leq x+L}\},\quad s=\sum\limits_{k+1}^n{(1-Q_k(l_k)})l_k^2$$Theorem 1. If$L>\max l_2$, then$$Q(L)\leq\frac{CL}{l\sqrt s},$$where$C$is an absolute constant.Special cases of this theorem correspond to the results of [1]–[4].