On the rate of approach of the distributions of sums of independent random variables to accompanying distributions

Let $\mathcal{F}$ be the set of all distribution functions on $R,\mathcal{F}^*$ the subset of $\mathcal{F}$ corresponding to symmetric random variables, $F^n$ $n$-times convolution of $F$ with itself, $E_a$ the distribution function corresponding to the unit mass at $a$, $|F-G|=\sup_x |F(x)-G(x)|$ for $F,G\in\mathcal{F}$.
It is proved that
$$ \frac{c_0}{n^{1/3}}<\sup_{F\in\mathcal{F}}\inf_a |(E_a F)^n-\exp\{n(E_a F-E_0)\}|\leq\frac{8}{n^{1/3}}, $$

$$ \frac{c_1}{\sqrt{n}}<\sup_{F\in\mathcal{F}^*}|F^n-\exp\{n(F-E_0)\}|<c_2\sqrt{\frac{\log n}{n}}. $$
Here the first right-hand inequality is Kolmogorov's uniform limit theorem in Le Cam's version.
We study also the closeness of distribution functions $\prod_i F_i E_{a_i}$ and $\exp\sum_i (F_iE_{a_i}-E_0)$ in the Kolmogorov–Smirnov and Lévy metrices.