Precise Rates in Log Laws for NA Sequences

Let X1,X2,… be a strictly stationary sequence of negatively associated (NA) random variables with EX1=0, set Sn=X1+⋯+Xn, suppose that σ2=EX12+2∑n=2∞EX1Xn>0 and EX12<∞, if −1<α≤1; EX12(log|X1|)α<∞, if α>1. We prove limε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, where κn=O(1/logn) and N is the standard normal random variable