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

Abstract

Let $0<p<2$. Let $\{X,X_n;n\ge 1\}$ be a sequence of independent and identically distributed $\mathbf{B}$-valued random variables and set $S_n={\sum }_{i=1}^n{X}_i,n\ge 1$. In this paper, a supplement to the classical laws of large numbers and the classical large deviations is provided. We show that if $S_n/n^{1/p}{\to }_{\mathbb{P}}0$, 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.

19 pages