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

Let us consider independent identically distributed random variables $X_1, X_2, \ldots\,$, 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 $\alpha\in (0, 2)$.
$$ \sum_n f_nP\{|U_n|\geq\varepsilon\varphi_n\}\sim \sum_n f_nP\{|\xi_\alpha|\ge\varepsilon\varphi_n\},\qquad\varepsilon\searrow 0, $$
Our basic purpose is to find conditions under which 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 $.