  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.
