Extended classical asymptotic expansions in the case of Gaussian limit distribution (Q1873208)

Let \(F_{n}\) denote the distribution of the normalized sum of iid radom variables \(X_{i}\) with common density \(cx^{-\alpha -1}\ln ^{\gamma }(x),\) \(x>r>1,\alpha \geq 2.\) Let the integer \(k\) be defined by \(E|X_{i}|^{k}<\infty \) and \(E|X_{i}|^{k+1}=\infty .\) The main result is an asymptotic expansion of \(F_{n}\) that consists of two terms. The first one is the classical Edgeworth expansion of order \(k\) whereas the second term includes \(\alpha \) and \(\gamma.\) The order of the remainder is \(n^{-(\alpha -2)/2}\ln ^{\gamma -1}(n)\) for \(\alpha \notin \mathbb{N}\) and \(n^{-(\alpha -2)/2}\ln ^{\gamma }(n)\) for \(\alpha \in \mathbb {N}\).