Some Estimates for the Maximum Cumulative Sum of Independent Random Variables

Let $S_n=\sum_{k=1}^n X_k$, $\overline{S}_n=\max_{1\ge k\le n} S_k$; $B_n^2=\sum_{k=1}^n \mathbf{D}X_k$,
$$ G(x)=\begin{cases} \sqrt{\frac{2}{\pi}}\int_0^x e^{-t^2/2}dt &(x\ge 0)\\ 0 &(x<0) \end{cases}, \quad L_{n,p}=\frac{\sum_{k=1}^n \mathbf{E}|X_k|^p}{B_n^p} \quad (p>2). $$
A sequence of independent symmetric random variables $\{X_n\}$ is constructed for which the estimste
$$ \sup_x|\mathbf{P}\{\overline{S}_n<xB_n\}-G(x)|=o(L_{n,p}^{1/p}) $$
ails to hold.