Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let ({X_i, ige 1}) be i.i.d. (mathbb {R}^d)-valued random vectors attracted to operator semi-stable laws and write (S_n=sum _{i=1}^{n}X_i). This paper investigates precise large deviations for both the partial sums (S_n) and the random sums (S_{N(t)}), where N(t) is a counting process independent of the sequence ({X_i, ige 1}). In particular, we show for all unit vectors (theta ) the asymptotics

$$begin{aligned} {mathbb P}(|langle S_n,theta rangle |>x)sim n{mathbb P}(|langle X,theta rangle |>x) end{aligned}$$

which holds uniformly for x-region ([gamma _n, infty )), where (langle cdot , cdot rangle ) is the standard inner product on (mathbb {R}^d) and ({gamma _n}) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and (mathbb {R}^d)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.