Geodesic
On a~local limit theorem for the sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 393-395

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Let $X_i$, $i\to\overline{1,\infty}$, be independent identically distributed random variables with $\mathbf EX_i=0$, $\mathbf DX_i=\sigma^2\infty$, and let $\displaystyle S_n=\sum_1^nX_i$, $\displaystyle\overline S_n=\max_{1\le k\le n}S_k$. A local limit theorem for the probabilities $\mathbf P(\overline S_n=x)$ is formulated in the case when $x=o(\sqrt n)$. This result complements the local limit theorem proved in [1] 
@article{TVP_1976_21_2_a14,
     author = {S. V. Nagaev and M. S. \`Eppel'},
     title = {On a~local limit theorem for the sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {393--395},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a14/}
}
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S. V. Nagaev; M. S. Èppel'. On a~local limit theorem for the sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 2, pp. 393-395. http://geodesic.mathdoc.fr/item/TVP_1976_21_2_a14/