Some results on probabilities of moderate deviations

TL;DR

This paper derives precise asymptotic estimates for moderate deviation probabilities of sums of i.i.d. random variables, revealing dependence on both variance and tail behavior on a specific scale involving a regularly varying function.

Contribution

It provides new moderate deviation results that incorporate both variance and tail distribution asymptotics, extending existing literature.

Findings

Asymptotic estimates depend on variance and tail behavior.

Results are on the scale of g(log n) for a regularly varying function g.

Provides comprehensive moderate deviation probabilities for all x > 0.

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 $X$.