On a Uniform Limit Theorem of A. N. Kolmogorov

Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent identically distributed random variables. Put $F(x)=\mathbf P\left\{{\xi_j<x}\right\}$, $F^n(x)=\mathbf P\left\{{\xi_1+\cdots+\xi_n<x} \right\}$ 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]: $\psi(n)< C''n^{-1/5}.$