arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2007.05150math.PR
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关于大数定律和大偏差的一个补充

A supplement to the laws of large numbers and the large deviations

Deli Li, Yu Miao

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19 pages
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摘要：

令$0 < p < 2$。设$\{X, X_{n}; n \geq 1\}$是独立同分布的$\mathbf{B}$值随机变量序列，并集合$S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$ .本文对经典大数定律和经典大偏差进行了补充。我们证明了，如果$S_{n}/n^{1/p} \rightarrow_{\mathbb{P}} 0$，那么，对于所有的$s > 0$，\[ \limsup_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = - (\barβ - p)/p \]和\[ \liminf_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = -(\underlineβ - p)/p, \]，其中\[ \barβ = - \limsup_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t} ~~\mbox{and}~~\underlineβ = - \liminf_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t}. \]证明这个结果的主要工具是对称化技术和Hoffmann-Jørgensen（1974），de Acosta（1981），以及Ledoux和Talagrand（1991）分别建立的三个强大的不等式。作为这一结果的特例，Hu和Nyubrien（2004）的主要结果不仅得到了改进，而且得到了推广.

英文摘要：

Let $0 < p < 2$. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independent and identically distributed $\mathbf{B}$-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. In this paper, a supplement to the classical laws of large numbers and the classical large deviations is provided. We show that if $S_{n}/n^{1/p} \rightarrow_{\mathbb{P}} 0$, then, for all $s > 0$, \[ \limsup_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = - (\barβ - p)/p \] and \[ \liminf_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left\|S_{n} \right\| > s n^{1/p} \right) = -(\underlineβ - p)/p, \] where \[ \barβ = - \limsup_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t} ~~\mbox{and}~~\underlineβ = - \liminf_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log \|X\| > t)}{t}. \] The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-Jørgensen (1974), de Acosta (1981), and Ledoux and Talagrand (1991), respectively. As a special case of this result, the main results of Hu and Nyrhinen (2004) are not only improved, but also extended.

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