Some characteristics of $n$-times convolutions of distributions

Let $\mathfrak A$ be the set of symmetric distributions, $\mathfrak M$ – the set of distributions with the median zero, $F^n$ – $n$-times convolution of $F$ with itself, $E_0$ – the distribution corresponding to the unit mass at 0. For $F,G\in\mathfrak M$ let us define
\begin{gather*} |F-G|=\sup_x|f((-\infty,x])-G((-\infty,x])|,\\ \operatorname{exp}(n(F-E_0))=\sum_{s=0}^\infty e^{-n}\frac{n^s}{s!}F^s. \end{gather*}
We prove that
\begin{gather*} c_1n^{-1/2}\le\sup_{F\in\mathfrak M}|F^n-F^{n+1}|\le c_2n^{-1/2},\\ c_3n^{-1/2}\le\sup_{F\in\mathfrak A}|F^n-\operatorname{exp}(n(F-E_0))|\le c_4n^{-1/2}. \end{gather*}