On the complete convergence of moments in exact asymptotics under normal approximation

For the sums of the form $\overline I_s(\varepsilon) = \sum_{n\geqslant 1} n^{s-r/2}\mathbf{E}|S_n|^r\,\mathbf I[|S_n|\geqslant \varepsilon\,n^\gamma]$, where $S_n = X_1 +\dots + X_n$, $X_n$, $n\geqslant 1$, is a sequence of independent and identically distributed random variables (r.v.'s) $s+1 \geqslant 0$, $r\geqslant 0$, $\gamma>1/2$, and $\varepsilon>0$, new results on their behavior are provided. As an example, we obtain the following generalization of Heyde's result [J. Appl. Probab., 12 (1975), pp. 173–175]: for any $r\geqslant 0$, $\lim_{\varepsilon\searrow 0}\varepsilon^{2}\sum_{n\geqslant 1} n^{-r/2} \mathbf{E}|S_n|^r\,\mathbf I[|S_n|\geqslant \varepsilon\, n] =\mathbf{E} |\xi|^{r+2}$ if and only if $\mathbf{E} X=0$ and $\mathbf{E} X^2=1$, and also $\mathbf{E}|X|^{2+r/2}<\infty$ if $r < 4$, $\mathbf{E}|X|^r<\infty$ if $r>4$, and $\mathbf{E} X^4 \ln{(1+|X|)}<\infty$ if $r=4$. Here, $\xi$ is a standard Gaussian r.v.