On the distribution of the maximum of cumulative sums of independent random variables

Let $X_1,\dots,X_n$ be independent random variables, $S_k=\sum_{j=1}^kX_j$, $\overline S_n=\max\limits_{1\le k\le n}S_k$. Set
$$ G(x)= \begin{cases} \sqrt{\frac2\pi}\int_0^xe^{t^2/2}\,dt,&x>0, \\ 0,&x\le0. \end{cases} $$
An estimate for $\sup|\mathbf P(\overline S_n<bx)-G(x)|$, where $b$ is an arbitrary positive number, is obtained without assumptions about the existence of moments. Some corrolaries are derived from this result. For example, if $\mathbf EX_k=0$ for all $k$ and $q_n^2=\sum_{k=1}^n\mathbf EX_k^2<\infty$, then
$$ \sup_x|\mathbf P(\overline S_n<q_nx)-G(x)|<\frac{\Lambda_n(\varepsilon)}{\varepsilon^2}+12\varepsilon $$
for any $\varepsilon>0$. Here $\Lambda_n(\varepsilon)$ is the Lindeberg ratio defined by (10).