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

Abstract

Let {X i , i ≥ 1} be i.i.d. R d -valued random vectors attracted to operator semi-stable laws and write S n = n i=1 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 , i ≥ 1}. In particular, we show for all unit vectors θ the asymptoticswhich holds uniformly for x-region [γ n , ∞), where ·, · is the standard inner product on R d and {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 R d -valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

Order of approximation in the central limit theorem for associated random variables and a moderate deviation result

“…Further, large deviation probability and moderate deviation probability investigations received much attention due to their importance in statistical inference and applied probability. We refer to monographs by Vardhan (1984), Dembo and Zeitouni (1998) and Hollander (2000) and recent papers by Wang (2015) andÇ aǧin et al (2016) for other references. These investigations are also useful in the construction of certain counter examples (see, for example, Tikhomirov, 1979 andBirkel, 1988).…”