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Let {Xi；i≥1} be a strictly stationary sequence of associated random variables with mean zero and let σ^2 =EX1^2+ 2∑j=2^∞EX1Xj with 0<σ^2<∞，Set Sm=∑i=1^n Xi, the precise asymptoties for ∑n≥1 n^r/p=2P(|Sn|≥εn1/p),∑n≥1 1/nP(|Sn|≥εn1/p)and ∑n≥1 (logn)^δ