Geodesic
On the distribution of the maximum of cumulative sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 708-715 
V. B. Nevzorov; V. V. Petrov. On the distribution of the maximum of cumulative sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 708-715. http://geodesic.mathdoc.fr/item/TVP_1969_14_4_a8/
@article{TVP_1969_14_4_a8,
     author = {V. B. Nevzorov and V. V. Petrov},
     title = {On the distribution of the maximum of cumulative sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {708--715},
     year = {1969},
     volume = {14},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_4_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $X_1,\dots,X_n$ be independent random variables, $S_k=\sum_{j=1}^kX_j$, $\overline S_n=\max\limits_{1\le k\le n}S_k$. Set $$ G(x)= \begin{cases} \sqrt{\frac2\pi}\int_0^xe^{t^2/2}\,dt,&x>0, \\ 0,&x\le0. \end{cases} $$ An estimate for $\sup|\mathbf P(\overline S_n, where $b$ is an arbitrary positive number, is obtained without assumptions about the existence of moments. Some corrolaries are derived from this result. For example, if $\mathbf EX_k=0$ for all $k$ and $q_n^2=\sum_{k=1}^n\mathbf EX_k^2<\infty$, then $$ \sup_x|\mathbf P(\overline S_n<q_nx)-G(x)|<\frac{\Lambda_n(\varepsilon)}{\varepsilon^2}+12\varepsilon $$ for any $\varepsilon>0$. Here $\Lambda_n(\varepsilon)$ is the Lindeberg ratio defined by (10).