Precise rates in the law of the logarithm for negatively associated random variables

Let {Xn;n≥1}{Xn;n≥1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set Sn=∑k=1nXk, Mn=maxk≤n|Sk|,n≥1Mn=maxk≤n|Sk|,n≥1. Suppose σ2=EX12+2∑k=2∞EX1Xk. We study the precise rates of a kind of weighted infinite series of P{Mn≥εσnlogn} and P{|Sn|≥εσnlogn} as ε↘0ε↘0, and P{Mn≤εσπ2n8logn} as ε↗∞ε↗∞. The results are related to the convergence rates of the law of the logarithm and the Chung type law of the logarithm.