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

Let 0 < p < 2. Let {X, X-n; n >= 1} be a sequence of independent and identically distributed B-valued random variables and set S-n = Sigma(n)(i=1) X-i, n >= 1. In this paper, an analogue of large deviation principle is established under assumption S-n/n(1/p) -> P 0 only. The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-Jorgensen [Sums of independent Banach space valued random variables, Studia Math. 52 (1974), pp. 159-186], de Acosta [Inequalities for B-valued random vectors with applications to the law of large numbers, Ann. Probab. 9 (1981), pp. 157-161] and Ledoux and Talagrand [Probability in Banach Spaces: Isoperimetry and Processes, Springer-Verlag, Berlin, 1991], respectively. As a special case of this result, the main results of Hu and Nyrhinen [Large deviations view points for heavy-tailedrandomwalks, J. Theoret. Probab. 17 (2004), pp. 761-768] are not only improved, but also extended.