Some results on probabilities of moderate deviations

Abstract

Let $\{X,X_n;n\ge 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}{X}^2<\infty$. Let $S_n={\sum }_{i=1}^n{X}_i$, $n\ge 1$. Let $g(\cdot ):[0,\infty )\to [0,\infty )$ be a nondecreasing regularly varying function with index $\ddot{I}\ge 0$ and ${\lim }_{t\to \infty }g(t)=\infty$. Let $\hat{I}¼=\mathbb{E}X$ and $\ddot{I}{ƒ}^2=\mathbb{E}(X-\hat{I}¼{)}^2$. In this paper, on the scale $g(\log n)$, we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form $\log \mathbb{P}\left(S_n-n\hat{I}¼>x\sqrt{ng(\log n)}\right)$, $\log \mathbb{P}\left(S_n-n\hat{I}¼<-x\sqrt{ng(\log n)}\right)$, and $\log \mathbb{P}\left(\left|S_n-n\hat{I}¼\right|>x\sqrt{ng(\log n)}\right)$ 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 $X$.