Some results on probabilities of moderate deviations

Abstract

Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}X^{2} < \infty$. Let $S_{n} = \sum_{i=1}^{n} X_{i}$, $n \geq 1$. Let $g(\cdot): ~[0, \infty) \rightarrow [0, \infty)$ be a nondecreasing regularly varying function with index $\rho \geq 0$ and $\lim_{t \rightarrow \infty} g(t) = \infty$. 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 $\displaystyle \log \mathbb{P}\left(S_{n} - n \mu > x \sqrt{ng(\log n)} \right)$, $\displaystyle \log \mathbb{P}\left(S_{n} - n \mu < -x \sqrt{ng(\log n)} \right)$, and $\displaystyle \log \mathbb{P}\left(\left|S_{n} - n \mu \right| > x \sqrt{ng(\log n)} \right)$ 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 $X$.