Geodesic
On the convergence speed of distribution of maximum sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 320-325

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Let $\xi_n$, $n=1,2,\dots,$ be a sequence of independent identically distributed random variables with $\mathbf M\xi=0$. Put $\sigma^2=\mathbf D\xi_1$, $c_3=\mathbf M|\xi_1|^3$, $S_n=\sum_{i=1}^n\xi_i$, $S_n^-=\max\limits_{1\le i\le n}S_i$, $\overline F_n(x)=\mathbf P(\overline S_n$. The following estimate is obtained: there exists an absolute constant $K$ such that $$ \sup_{0\le x\infty}|\overline F_n(x\sigma\sqrt n)-\biggl(\frac2\pi\biggr)^{1/2}\int_0^xe^{-u^{2/3}}\,du|\frac{c_3^2}{\sigma^6\sqrt n}. $$ 
@article{TVP_1970_15_2_a11,
     author = {S. V. Nagaev},
     title = {On the convergence speed of distribution of maximum sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {320--325},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1970_15_2_a11/}
}
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S. V. Nagaev. On the convergence speed of distribution of maximum sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 320-325. http://geodesic.mathdoc.fr/item/TVP_1970_15_2_a11/