The behaviour of sums of independent random variables

In the paper, a necessary and sufficient condition is given in order that
$$ -\infty<\varliminf_{n\to\infty}\frac{S_n-mS_n}{a_n}\le\varlimsup\frac{S_n-mS_n}{a_n}<\infty, $$
where $\{\xi_n\}$ is a sequence of independent random variables, $S_n=\xi_1+\dots+\xi_n$; $m\xi$ is the median of $\xi$; $\{a_n\}$ is an increasing sequence of positive numbers such that there exists, a sequence of indices $\{m_n\}$ for which
$$ 1<C_1\le\frac{a_{m_{n+1}}}{a_{m_n}}\le C_2<\infty. $$