On the convergence rate in “the exact asymptotics” for random variables with a stable distribution

In the note we present the conditions under which relations of the type
$$ \lim\limits_{\varepsilon\searrow 0}\left(\sum\limits_{n\ge 1} r(n) \mathbf{P}(Y_\alpha\ge f(\varepsilon g(n))) - \nu(\varepsilon) \right) = C $$
hold, where a random variable $Y_\alpha$ has a stable distribution, $C$ is a constant, and non-negative functions $r$, $f$ and $g$ satisfy certain conditions. The obtained results allow to make more precise and to complement results, related to the convergence rate in the so called “exact asymptotics”.