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Teor. Veroyatnost. i Primenen., 1960, Volume 5, Issue 1, Pages 103–113 (Mi tvp4816)  

Short Communications

On a Uniform Limit Theorem of A. N. Kolmogorov

Yu. V. Prokhorov

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: Let $\xi_1,\xi_2,…,\xi_n,…$ be a sequence of independent identically distributed random variables. Put $F(x)=\mathbf P\{{\xi_j<x}\}$, $F^n(x)=\mathbf P\{{\xi_1+\cdots+\xi_n<x} \}$ and
$$\psi(n)=\sup\limits_f\inf\limits_{G\in\mathfrak G}\sup\limits_x|{F^n(x)-G(x)}|,$$
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 Kolmogorovs estimate [8]: $\psi(n)< C"n^{-1/5}.$

Full text: PDF file (884 kB)

English version:
Theory of Probability and its Applications, 1960, 5:1, 98–106

Received: 16.10.1959

Citation: Yu. V. Prokhorov, “On a Uniform Limit Theorem of A. N. Kolmogorov”, Teor. Veroyatnost. i Primenen., 5:1 (1960), 103–113; Theory Probab. Appl., 5:1 (1960), 98–106

Citation in format AMSBIB
\Bibitem{Pro60}
\by Yu.~V.~Prokhorov
\paper On a Uniform Limit Theorem of A. N. Kolmogorov
\jour Teor. Veroyatnost. i Primenen.
\yr 1960
\vol 5
\issue 1
\pages 103--113
\mathnet{http://mi.mathnet.ru/tvp4816}
\transl
\jour Theory Probab. Appl.
\yr 1960
\vol 5
\issue 1
\pages 98--106
\crossref{https://doi.org/10.1137/1105008}


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