Geodesic
On a Uniform Limit Theorem of A. N. Kolmogorov
Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 103-113

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Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent identically distributed random variables. Put $F(x)=\mathbf P\left\{{\xi_j$, $F^n(x)=\mathbf P\left\{{\xi_1+\cdots+\xi_n$ and $$\psi(n)=\sup\limits_f\inf\limits_{G\in\mathfrak G}\sup\limits_x\left|{F^n(x)-G(x)}\right|,$$ where $\mathfrak{G}$ is a set of all infinitely divisible laws. Then, there exist two absolute constants $C'$ and $C''$ such that $$C'n^{-1}(\log n)^{-1}\psi(n) C''n^{-1/3}(\log n )^2.$$ The right-hand inequality $(*)$ is an improvement of Kolmogorov’s estimate [8]: $\psi(n) C''n^{-1/5}.$ 
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     author = {Yu. V. Prokhorov},
     title = {On a {Uniform} {Limit} {Theorem} of {A.} {N.} {Kolmogorov}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {103--113},
     publisher = {mathdoc},
     volume = {5},
     number = {1},
     year = {1960},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a7/}
}
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Yu. V. Prokhorov. On a Uniform Limit Theorem of A. N. Kolmogorov. Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 1, pp. 103-113. http://geodesic.mathdoc.fr/item/TVP_1960_5_1_a7/