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

Let 0<p<2. Let {X,Xn;n≥ 1} be a sequence of independent and identically distributed B-valued random variables and set Sn=∑i=1nXi, n≥ 1. In this paper, an analogue of large deviation principle is established under assumption Sn/n1/p→P0 only. The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-Jørgensen [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-tailed random walks, J. Theoret. Probab. 17 (2004), pp. 761–768] are not only improved, but also extended.