On the existence of exact upper sequences

The following results are obtained.
Theorem 2. Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent random variables and
$$ \frac{z^2\mathbf P\{|\xi_n-\mu(\xi_n)|>z\}}{\int_{|x|\le z}x^2\,d\mathbf P\{\xi_n-\mu(\xi_n)<x\}}\ge c>0, $$
$n=1,2,\dots$, then there exists no sequence $a_1,a_2,\dots,a_n,\dots$, $a_n\uparrow\infty$ as $n\to\infty$, having the property
$$ \mathbf P\biggl\{\varlimsup_{m\to\infty}\frac{|S_n-\mu(S_n)|}{a_n}=1\biggr\}=1,\eqno(*) $$
where $S_n=\sum_{k=1}^n\xi_k$ and $\mu(\eta)$ is the median of $\eta$.
Theorem 4. Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be a sequence of independent equally distributed random variables, then there exists no sequence $a_1,a_2,\dots,a_n,\dots$ with the properties $(*)$ and
$$ \sum_{k=n}^\infty a_k^{-2}\le Cna_n^{-2} $$
for all $n$ and $C>0$.
In the end of the paper an example is constructed which gives the negative answer to the question stated in [1].