Some results on probabilities of moderate deviations

Let $\{X,X_n;n\ge 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}{X}^2<\mathrm{\infty }$. Let $S_n={\sum }_{i=1}^n{X}_i$, $n\ge 1$. Let $g(\cdot ):[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a nondecreasing regularly varying function with index $\rho \ge 0$ and ${\lim }_{t\to \mathrm{\infty }}g(t)=\mathrm{\infty }$. 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 ${\displaystyle \log \mathbb{P}(S_n-n\mu >x\sqrt{ng(\log n)})}$, ${\displaystyle \log \mathbb{P}(S_n-n\mu <-x\sqrt{ng(\log n)})}$, and ${\displaystyle \log \mathbb{P}(\mid S_n-n\mu \mid >x\sqrt{ng(\log n)})}$ 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 $X$.