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

Let {X-n; n >= 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S-n =Sigma(n)(k=1) X-k , M-n = max(k <= n) vertical bar S-k vertical bar, n >= 1. Suppose sigma(2) = EX12 + 2 Sigma(infinity)(k=2) EX1 X-k. We study the precise rates of a kind of weighted infinite series of P{M-n >= epsilon sigma root n log n} and P{vertical bar S-n vertical bar >= epsilon sigma root n log n} as epsilon SE arrow 0, and P{M-n <= epsilon sigma root pi(2)n/8 log n} as epsilon NE arrow infinity. The results are related to the convergence rates of the law of the logarithm and the Chung type law of the logarithm. (c) 2007 Elsevier Ltd. All rights reserved.