On Evaluating the Concentration Functions

Let $\xi_1,\dots,\xi_n$ be independent random variables,
$$Q_k\{l\}=\mathop{\sup}\limits_x\mathbf P\{x\leq\xi _k\leq x+l\},\\Q(L)=\mathop{\sup}\limits_x\mathbf P\{{x\leq\xi_1+\cdots+\xi_n\leq x+L}\},\quad s=\sum\limits_{k+1}^n(1-Q_k(l)).$$

Theorem 1. If $L\ge l$, then
$$Q(L)\leq\frac{CL}{l\sqrt s},$$
where $C$ is an absolute constant. This is a refinement of the main theorem in [1].