On a weak form of a local limit theorem

Let $\{\xi_{nk}\}$ be a sequence of series of independent random variables. In the paper, a limit theorem for
$$ \mathbf P\biggl\{\sum_{k=1}^n\xi_{nk}\in\biggl[x-\frac{\lambda_n}{2},x+\frac{\lambda_n}{2}\biggr]\biggr\}, $$
where $\displaystyle\lambda_n=o\biggl(\sum_{k=1}^n\mathbf D\xi_{nk}\biggr)$, is obtained