Geodesic
On asymptotic behaviour of the remainder term in the central limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 1, pp. 109-119 
L. V. Rozovskiǐ. On asymptotic behaviour of the remainder term in the central limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 1, pp. 109-119. http://geodesic.mathdoc.fr/item/TVP_1978_23_1_a7/
@article{TVP_1978_23_1_a7,
     author = {L. V. Rozovskiǐ},
     title = {On asymptotic behaviour of the remainder term in the central limit theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {109--119},
     year = {1978},
     volume = {23},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1978_23_1_a7/}
}
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Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with zero means and unit variances. Let $k_n$ be a sequence of natural numbers, $k_n\to\infty$, $k_{n+1}/k_n\to 1$ ($n\to\infty$), $$ F_n(x)=\mathbf P\biggl\{\frac{1}{\sqrt{k_n}}\sum_{k=1}^{k_n}X_k<x\biggr\},\qquad \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt. $$ We study conditions under which $$ F_n(x)=\Phi(x)+\frac{\Psi(x)+o(1)}{\mu_n}\qquad (n\to\infty) $$ uniformly in $x$, $-\infty, where $\mu_n$ is a positive sequence such that $\mu_n\to\infty$, $\mu_n=o(k_n)$ ($n\to\infty$).