Some results on probabilities of moderate deviations
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866918005538029568 |
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| author | Li, Deli Miao, Yu Qi, Yongcheng |
| author_facet | Li, Deli Miao, Yu Qi, Yongcheng |
| contents | 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 $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_03545 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Some results on probabilities of moderate deviations Li, Deli Miao, Yu Qi, Yongcheng Probability Primary: 60F10, Secondary 60B12, 60F05, 60G50 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 $X$. |
| title | Some results on probabilities of moderate deviations |
| topic | Probability Primary: 60F10, Secondary 60B12, 60F05, 60G50 |
| url | https://arxiv.org/abs/2207.03545 |