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

Let $X_1,\dots,X_n$ be independent random variables with $\mathbf EX_k=0$ and $\mathbf E|x_k|^3=\gamma_{3k}<\infty$. Let
$$ S_0=0,\quad S_k=\sum_{i=1}^kX_i\quad(k=1,\dots,n),\quad\overline{S_n}=\max\limits_{0\le k\le n}S_k,\quad B_k^2=\sum_{i=1}^k\mathbf DX_i. $$
In the paper, some bounds for
$$ \Delta_n(x)=\mathbf P\{\overline{S_n}<x\}-\sqrt{\frac2\pi}\int_0^{x/B_n}e^{-y^2/2}\,dy\quad(x\ge0) $$
are obtained. The main result is the following
Theorem. {\em Let $x\ge0$. Then
$$ |\Delta_n(x)|\le C\sum_{k=1}^n\frac{x+\rho_k}{x+\rho_k+B_k}\cdot\frac{B_n\gamma_{3k}}{(B_k^2+x^2)(B_n+x)B_{k-1,n}} $$
where $\rho_k=\max\limits_{i\le k}\gamma_{3i}/\mathbf DX_i$ and} $B_{k-1,n}=(\sum_{i=k}^n\mathbf DX_i)^{1/2}$.