Tail behavior of random sums of negatively associated increments

Let { X , X k , k ⩾ 1 } be a sequence of negatively associated random variables with a common distribution function and finite expectation and let τ be a nonnegative integer-valued random variable independent of { X k , k ⩾ 1 } . In this paper we give unified form for the asymptotic behavior of the random sums S τ = ∑ k = 1 τ X k in the case of lim x → + ∞ P ( X > x ) P ( τ > x ) = C ∈ [ 0 , + ∞ ] . The results extend the earlier results of Aleškevičiené et al. (2008) [2].