Summary: 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 C. C. Heyde’s result [J. Appl. Probab. 12, 173–175 (1975; Zbl 0305.60008)]: 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.