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

Abstract

Let 0<p<2. Let be a sequence of independent and identically distributed -valued random variables and set . In this paper, an analogue of large deviation principle is established under assumption 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.