论文标题
某些结果是关于中等偏差的概率
Some results on probabilities of moderate deviations
论文作者
论文摘要
令$ \ {x，x_ {n}; n \ geq 1 \} $是i.i.d的序列。具有$ \ mathbb {e} x^{2} <\ infty $的非脱位实价随机变量。令$ s_ {n} = \ sum_ {i = 1}^{n} x_ {i} $，$ n \ geq 1 $。令$ g（\ cdot）：〜[0，\ infty）\ rightArrow [0，\ infty）$是一种不定期变化的功能，使用索引$ρ\ geq 0 $和$ \ lim_ {t \ rightArrow \ rightArrow \ rightarrow \ infty \ infty} g（t）g（t）= \ infty $。令$μ= \ mathbb {e} x $和$σ^{2} = \ mathbb {e}（x -μ）^{2} $。在本文中，在规模$ g（\ log n）$上，我们获得了精确的渐近估计值，以$ \ displaystyle \ log \ log \ mathbb {p} \ left（s_ {n}-nμ> x \ x \ sqrt { \ Mathbb {p} \ left（s_ {n} - nμ<-x \ sqrt {ng（\ log n）} \ right）$，和$ \ displayStyle \ log \ log \ log \ log \ mathbb {p} \ left（ \ right）$ for hast $ x> 0 $。与文献中已知的结果不同，本文确定的中等偏差结果取决于$ x $的尾巴分布的方差和渐近行为。
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 $X$.
