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

Abstract: 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) = - (\bar{\beta} - p)/p ] and [ \liminf_{n \to \infty} \frac{1}{\log n} \log \mathbb{P}\left(\left|S_{n} \right| > s n^{1/p} \right) = -(\underline{\beta} - p)/p, ] where [ \bar{\beta} = - \limsup_{t \rightarrow \infty} \frac{\log \mathbb{P}(\log |X| > t)}{t} \mbox{and}\underline{\beta} = - \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-J{\o}rgensen (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.