Let {εt; t ∈ Z + } be a strictly stationary sequence of associated random variables with mean zeros, let 0 < Eε1 2 < ∞ and σ 2 = Eε1 2 +2Xt, n ≥ 1. Assume that E|ε1| 2+δ < ∞ for some δ > 0 and μ(n) = O(n −ρ ) for some ρ > 0. This paper achieves a general law of precise asymptotics for {Sn}. §1 Introduction
and ψ(x) and H(x)be the positive functions defined on [1, +∞), and P (ψ, H, ε) = ∞ n=1 ψ(n)P (|S n | > εH(n)), ∀ε > 0, we call ψ(x) and H(x) weighted function and boundary function respectively.A finite collection of random variables ε 1 , ε 2 , · · · , ε m is said to be associated if Cov{f (ε 1 , · · · , ε m ), g(ε 1 , · · · , ε m )} ≥ 0 for any two coordinatewise nondecreasing functions f, g on R m such that the covariance is defined. An infinite collection is associated if every finite subcollection is associated. Associated random variables were introduced by Esary, et al. [1] and have found many applications in reliability theory. Many authors have studied this concept and proved interesting results and applications, we refer to Newman [2] for the central limit theorem, Newman and Wright [3] for the invariance principle, Dabrowski [4] for the functional law of the iterated logarithm, Birkel [5] and Lin [6] for the moment bounds, Birkel [7] for the strong law of large numbers, Yang [8] for the complete convergence, Mi [9] for the precise asymptotics in the Baum-Katz and Davis law of large numbers for special weighted functions and boundary functions.