Comparison theorems for small deviations of weighted series

We study comparison theorems for small deviation probabilities of weighted series and obtain more refined versions of the previous results by the theme. In particular, we prove the following result.
Theorem. Let a positive random variable $X$ belong to the domain of attraction of a stable law with an index more than 1 and let its distribution function be regularly varying at zero with an exponent $\beta>0$. If $\{X_n\}_{n\ge 1}$ are independent copies of $ X$, and $\{a_n\}$ and $\{b_n\}$ are positive summable sequences such that $ \sum\limits_{n\ge 1} |1-a_n/b_n|<\infty,$ then as $r\to 0^+$
$$ \mathbb{ P}\Big(\sum\limits_{n\ge 1} a_n\,X_n < r\Big)\sim \Big(\prod\limits_{n\ge 1} b_n/a_n\Big)^\beta\,\mathbb{P}\Big(\sum\limits_{n\ge 1} b_n\,X_n < r\Big). $$