Summary: Let \(\{X,X_i: i\geq 1\}\) be a sequence of i.i.d. random variables with common distribution function \(F(x)\) and let \(S_n= \sum_{i=1}^n X_i\), \(n\geq 1\). Suppose that \(F\) belongs to the domain of attraction of a nondegenerate stable distribution \(G\) with characteristic exponent \(\alpha\) \((0<\alpha\leq 2)\), i.e., that \((S_n-a_n)/b_n@>D>>G\), as \(n\to\infty\) for suitable \(a_n\) and \(b_n\). Let \(g(x)\) (defined on \([0,+\infty)\)) be an increasing functions such that \(g^{-1}(x)\) is a regular varying function with index \(\rho\) \((0\leq\rho<\alpha)\). We prove that
\[ \lim_{\varepsilon\searrow 0} \frac{1}{g^{-1}(1/\varepsilon)} \sum_{n=n_0}^\infty h'(n) \text{Pr}(|S_n-a_n|> \varepsilon b_ng(h(n)))= E|Z|^p, \]
for some positive integer \(n_0\) and nonnegative function \(h(x)\), where \(Z\) is a random variable having the distribution \(G\).