Geodesic
Some Estimates for the Maximum Cumulative Sum of Independent Random Variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 2, pp. 402-405

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Let $S_n=\sum_{k=1}^n X_k$, $\overline{S}_n=\max_{1\ge k\le n} S_k$; $B_n^2=\sum_{k=1}^n \mathbf{D}X_k$, $$ G(x)=\begin{cases} \sqrt{\frac{2}{\pi}}\int_0^x e^{-t^2/2}dt (x\ge 0)\\ 0 (x0) \end{cases}, \quad L_{n,p}=\frac{\sum_{k=1}^n \mathbf{E}|X_k|^p}{B_n^p} \quad (p>2). $$ A sequence of independent symmetric random variables $\{X_n\}$ is constructed for which the estimste $$ \sup_x|\mathbf{P}\{\overline{S}_n\}-G(x)|=o(L_{n,p}^{1/p}) $$ ails to hold. 
@article{TVP_1973_18_2_a22,
     author = {T. V. Arak and V. B. Nevzorov},
     title = {Some {Estimates} for the {Maximum} {Cumulative} {Sum} of {Independent} {Random} {Variables}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {402--405},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_2_a22/}
}
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T. V. Arak; V. B. Nevzorov. Some Estimates for the Maximum Cumulative Sum of Independent Random Variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 2, pp. 402-405. http://geodesic.mathdoc.fr/item/TVP_1973_18_2_a22/