On a local limit theorem for the sums of independent random variables

Let $X_i$, $i\to\overline{1,\infty}$, be independent identically distributed random variables with $\mathbf EX_i=0$, $\mathbf DX_i=\sigma^2<\infty$, and let $\displaystyle S_n=\sum_1^nX_i$, $\displaystyle\overline S_n=\max_{1\le k\le n}S_k$. A local limit theorem for the probabilities $\mathbf P(\overline S_n=x)$ is formulated in the case when $x=o(\sqrt n)$. This result complements the local limit theorem proved in [1]