Let 0<p<2. Let $\{X,X_n;n\ge 1\}$ be a sequence of independent and identically distributed $\mathbf{B}$-valued random variables and set $S_n=\sum _{i=1}^n{X}_i,n\ge 1$. In this paper, an analogue of large deviation principle is established under assumption $S_n/n^{1/p}{\to }_{\mathbb{P}}0$ 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.

让 0 < p < 2。让$\{X,X_n;n\ge 1\}$ 是一个独立同分布的序列 $\mathbf{乙}$-valued 随机变量并设置 ${秒}_n=\sum _{\mathrm{一世}=1}^n{X}_{\mathrm{一世}},n\ge 1$. 本文在假设条件下建立了大偏差原理的类比${秒}_n/n^{1/磷}{\to }_{\mathbb{磷}}0$ 只要。证明这一结果的主要工具是对称化技术和 Hoffmann-Jørgensen 建立的三个强大的不等式 [独立 Banach 空间值随机变量的总和，Studia Math。52 (1974), pp. 159–186], de Acosta [B 值随机向量的不等式在大数定律中的应用，Ann。可能。9 (1981), pp. 157–161] 和 Ledoux 和 Talagrand [ Banach 空间中的概率：等周测量和过程，Springer-Verlag，柏林，1991]。作为这个结果的一个特例，Hu 和 Nyrhinen 的主要结果 [重尾随机游走的大偏差观点，J. Theoret 。可能。17 (2004), pp. 761–768] 不仅得到了改进，而且得到了扩展。