On a Uniform Limit Theorem of A. N. Kolmogorov
Yu. V. Prokhorov · Theory of Probability and Its Applications · 1960
Let $\xi _1 ,\xi _2 , \cdots ,\xi _n , \cdots $ be a sequence of independent identically distributed random variables. Put $F(x) = {\bf P}\left\{ {\xi _j < x} \right\},\quad,F^n (x) = {\bf P}\left\{ {\xi _1 + \cdots + \xi _n < x} \right\}$ and \[ \psi (n) = \mathop {\sup }\limits_f \mathop {\inf }\limits_{G \in \mathfrak{G}} \mathop {\sup }\limits_x \left| {F^n (x) - G(x)} \right| \] 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]: \[ \psi (n) < C''n^{ - 1/5}. \]