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

Let $$\{X_i, i\ge 1\}$$ { X i , i ≥ 1 } be i.i.d. $$\mathbb {R}^d$$ R d -valued random vectors attracted to operator semi-stable laws and write $$S_n=\sum _{i=1}^{n}X_i$$ S n = ∑ i = 1 n X i . This paper investigates precise large deviations for both the partial sums $$S_n$$ S n and the random sums $$S_{N(t)}$$ S N ( t ) , where N(t) is a counting process independent of the sequence $$\{X_i, i\ge 1\}$$ { X i , i ≥ 1 } . In particular, we show for all unit vectors $$\theta $$ θ the asymptotics $$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$ P ( | ⟨ S n , θ ⟩ | > x ) ∼ n P ( | ⟨ X , θ ⟩ | > x ) which holds uniformly for x-region $$[\gamma _n, \infty )$$ [ γ n , ∞ ) , where $$\langle \cdot , \cdot \rangle $$ ⟨ · , · ⟩ is the standard inner product on $$\mathbb {R}^d$$ R d and $$\{\gamma _n\}$$ { γ 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 $$\mathbb {R}^d$$ R d -valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.