On the probabilities of large deviations for the maximum of sums of independent random variables

Let $\xi_1,\xi_2,\dots$ be a sequence of independent identically distributed random variables with, non-degenerate distribution function $F(x)$,
$$a=\mathbf E\xi_1,\quad\sigma^2=\mathbf D\xi_1,\quad S_{n}=\sum_{l=1}^n\xi_l,\quad \overline S_n=\max_{1\le k\le n}S_k,\quad\overline F(x)=\mathbf P\{\bar S_n<x\}$$
. We prove that if $a=0$ and
$$\int_{-\infty}^{\infty} e^{hy}\,dF(y)< \infty,\qquad |h|\le A,\ A>0,$$
then for $n\to\infty$, $1<x=o(\sqrt{n})$
$$\frac{1-\overline F_n(x\overline{\sigma}\sqrt{n})}{1-G(x)}= \exp\biggl\{\frac{x^{3}}{\sqrt{n}}\lambda\biggl(\frac{x}{\sqrt{n}}\biggr)\biggr\} \biggl[1+O\biggl(\frac{x}{\sqrt{n}}+e^{-x^2/8}\biggr)\biggr],$$
where $\displaystyle G(x)=(2/\pi)^{1/2}\int_{0}^x e^{-u^2/2}\,du$ ($x\ge 0$), $G(x)=0$ ($x<0$) and $\lambda(u)$ is a Cramer's power series. Analogous statement is proved for the case $a>0$. We obtain also the theorems on the probabilities of large deviations for $\overline S_n$ in the Linnik's zones.