Limit theorems for large derivations of sums of independent random variables with common distribution function from the domain of attraction of the normal law

In the note some aspects of an asymptotic behavior of the probability $\mathbf P\bigl(S_n\ge\alpha b_n\bigr)$ are studied, where $S_n$ is sum of $n$ independent random variables with a common distribution function from the domain of attraction of a normal law, $\alpha$ is a positive number and $b_n$ is a non-decreasing sequence, which tends to infinity and satisfies some additional assumptions. In particular, we obtain the necessary and sufficient conditions under which the series $\sum\limits_n f_n\,\mathbf P\bigl( S_n\ge\alpha b_n \bigr)$ converges or, being properly normalized, has a limit if $\alpha\searrow\alpha_0$, where $\alpha_0$ is a positive constant and $f_n$ is some positive sequence of rather a general form.