On exact asymptotics in the weak law of large numbers for sums of independent random variables with a common distribution function from the domain of attraction of a stable law. II

Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that
$$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qquad weakly as\quad n\to\infty, $$
where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable $\xi_\alpha$ has a stable distribution with characteristic exponent $\alpha\in[1,2]$.
Our basic purpose is to find conditions under which
$$ \sum_n f_n{P}\big\{U_n\geq\varepsilon\varphi_n\big\}\sim \sum_n f_n{P}\big\{\xi_\alpha\ge\varepsilon\varphi_n\big\}, \qquad\varepsilon\searrow 0, $$
with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.