Geodesic
On the probabilities of moderate deviations for sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 858-865

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Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables and $\sigma>0$. Put $$ F_n(x)=\mathbf P\biggl\{\sum_{i=1}^nX_i\biggr\},\qquad\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-t^2/2}\,dt. $$ Necessary and sufficient conditions are found for the validity of the relation $$ 1-F_n(x\sigma\sqrt n)=(1-\Phi(x))(1+o(1)),\qquad 0\le x\le c\sqrt{\log n},\qquad n\to\infty. $$ 
@article{TVP_1979_24_4_a18,
     author = {N. N. Amosova},
     title = {On the probabilities of moderate deviations for sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {858--865},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a18/}
}
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N. N. Amosova. On the probabilities of moderate deviations for sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 4, pp. 858-865. http://geodesic.mathdoc.fr/item/TVP_1979_24_4_a18/