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Let $0 < p < 2$. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independentand 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 classicallaws of large numbers and the classical large deviations is provided. We showthat 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 thisresult are the symmetrization technique and three powerful inequalitiesestablished by Hoffmann-J{\o}rgensen (1974), de Acosta (1981), and Ledoux andTalagrand (1991), respectively. As a special case of this result, the mainresults of Hu and Nyrhinen (2004) are not only improved, but also extended.

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