论文标题
大量法律和大偏差的补充
A supplement to the laws of large numbers and the large deviations
论文作者
论文摘要
令$ 0 <p <2 $。令$ \ {x，x_ {n}; n \ geq 1 \} $是一系列独立且相同分布的$ \ mathbf {b} $ - 估值随机变量，并设置$ s_ {n} = \ sum_ {i = 1}^{n}^{n} x__ {i} x_ {i}，〜n \ geq 1 $。在本文中，提供了大量经典定律和经典大偏差的补充。我们表明，如果$ s_ {n}/n^{1/p} \ rightArrow _ {\ mathbb {p}} 0 $，那么，对于所有$ s> 0 $，\ [\ limsup_ {n \ to \ Mathbb {p} \ left（\ left \ | s_ {n} \ right \ |> s n^{1/p} \ right）= - （\barβ-p）/p \]和\ [\ liminf_ { \ Mathbb {p} \ left（\ left \ | s_ {n} \ right \ |> s n^{1/p} \ right）= - （\undeslineβ -p）/p，\ \ \ \ \ [\barβ= - barβ= - \ limsup_ { \ | x \ |。 \]证明这一结果的主要工具是霍夫曼·乔根森（Hoffmann-Jørgensen，1974），de Acosta（1981）和Ledoux and Talagrand（1991）建立的对称技术和三种强大的不平等现象。作为此结果的特殊情况，Hu和Nyrhinen（2004）的主要结果不仅得到了改善，而且还扩大了。
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.
