Let {X, Xn; n ≥ 1} be a sequence of i.i.d. non-degenerate real-valued random variables with \({\mathbb E}{X}^{2} < \infty\) . Let \(S_{n}=\sum\nolimits_{i=1}^{n} X_{i}\) , n ≥ 1. Let g(·): [0, ∞) → [0, ∞) be a nondecreasing regularly varying function with index ρ ≥ 0 and \(\lim\nolimits_{{t\rightarrow\infty}} g(t)=\infty\) . Let \(\mu = {\mathbb E}X\) and \({\sigma^{2}}={\mathbb E}{(X-\mu)}^{2}\) . In this paper, on the scale g(log n), we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form \(\log \, {\mathbb P}(S_{n} - {n}\mu > x{\sqrt {ng(\log \ n)})}\) , \(\log \, {\mathbb P}(S_{n} - {n}\mu < -x{\sqrt {ng(\log \ n)})}\) , and \(\log \, {\mathbb P}(\vert S_{n} - {n}\mu \vert > x{\sqrt {ng(\log \ n)})}\) for all x > 0. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of X.
