On complete convergence of moments of i.i.d.r.v. with finite variances

Let $\{X_n\}, n\ge 1,$ be a sequence of independent random variables with common distribution functions, zero means and unit variances, $\bar{S}_n =( X_1 +\cdots + X_n)/\sqrt n$. The main goal of this note is a study of the behavior of sums
$$ \Sigma_r(\varepsilon) = \sum\limits_{n\ge 1} n^s \mathbf{E} \bar S^r_n I[\bar S_n\ge \varepsilon n^\delta], $$
as $\varepsilon\to +0$ under optimal (that is, necessary) moment assumptions, where $\delta, s, r$ are some constants, such that $\delta> 0$ and $s+1$ and $r$ are non-negative. In particular, it is shown that if $s>-1/2$ and $(2-r) \delta = s+1$, then
$$ \varepsilon^{2-r} \Sigma_r(\varepsilon) = \dfrac{1}{2\delta (2-r)} + O \big(\lambda(\rho)\big),\ \rho=\varepsilon^{-1/2\delta}, \lambda(\rho)=\mathbf{E} X_1^2 \Big(1 \land \dfrac{| X_1|}{\rho}\Big). $$
A similar estimate with a more complicated formulation holds also in the case $-1<s\le -1/2$. Thus, for $\delta=1/2$ we generalize the pioneering result of Heyde (Appl. Probab., 1975) and most its refinements (e.g. due to He and Xie (Acta Math. Appl. Sin., 2013)), as well as the corresponding statements of Liu and Lin (Statist. Probab. Lett. 2006) and Kong and Dai (Stoch. Dynamics, 2017).