Some results on probabilities of moderate deviations
Deli Li, Yu Miao, Yongcheng Qi
TL;DR
The paper investigates moderate deviations for sums of i.i.d. real-valued variables with finite second moment on the scale $g(\log n)$, where $g$ is regularly varying with index $\rho\ge0$. It derives precise asymptotics for $\log \mathbb{P}(S_n - n\mu > x\sqrt{n g(\log n)})$, $\log \mathbb{P}(S_n - n\mu < -x\sqrt{n g(\log n)})$, and $\log \mathbb{P}(|S_n - n\mu| > x\sqrt{n g(\log n)})$, showing the rate is governed by the Gaussian term $x^{2}/(2\sigma^{2})$ and the tail decay via regularly varying parameters $\overline{\lambda}_1, \underline{\lambda}_1, \overline{\lambda}_2, \underline{\lambda}_2, \overline{\lambda}, \underline{\lambda}$, with the limits expressed as $-\big( x^{2}/(2\sigma^{2}) \wedge \lambda/2^{\rho} \big)$. The work provides both upper and lower bounds (Theorem 2.1) and a set of equivalences and necessary conditions (Theorem 2.2) for these moderate deviations under minimal moment assumptions, employing truncation, conditional probability arguments, and Kolmogorov exponential inequalities. The results extend classical moderate deviation theory by incorporating tail behavior alongside variance, highlighting how tail regular variation shapes rare-event probabilities in sums.
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 $ρ\geq 0$ and $\lim_{t \rightarrow \infty} g(t) = \infty$. Let $μ= \mathbb{E}X$ and $σ^{2} = \mathbb{E}(X - μ)^{2}$. In this paper, on the scale $g(\log n)$, we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form $\displaystyle \log \mathbb{P}\left(S_{n} - n μ> x \sqrt{ng(\log n)} \right)$, $\displaystyle \log \mathbb{P}\left(S_{n} - n μ< -x \sqrt{ng(\log n)} \right)$, and $\displaystyle \log \mathbb{P}\left(\left|S_{n} - n μ\right| > x \sqrt{ng(\log n)} \right)$ 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 $X$.