Geodesic
On the approximation by the accompanying laws of $n$-fold convolutions
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 225-246

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Let $F$ be a probability distribution on $R$ having nonnegative characteristic function and let $E$ be the distribution with the unit mass at the origin. It is proved that $$ \sup_x|F^n([x,x+h))-e^{n(F-E)}([x,x+h))| \le C\gamma_h^{1/3}(|{\ln\gamma_h}|+1)^{13/3}n^{-1} $$ for any natural number $n$ and $h>0$. Here $C$ is an absolute constant and $\gamma_h$ denotes the value of the concentration function of the distribution $e^{n(F-E)}$ at the point $h$. 
@article{TVP_1980_25_2_a0,
     author = {T. V. Arak},
     title = {On the approximation by the accompanying laws of $n$-fold convolutions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {225--246},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {1980},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a0/}
}
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T. V. Arak. On the approximation by the accompanying laws of $n$-fold convolutions. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 225-246. http://geodesic.mathdoc.fr/item/TVP_1980_25_2_a0/