Maximal inequalities and a law of the iterated logarithm for negatively associated random fields

The exponential inequality of the maximum partial sums is a key to establish the law of the iterated logarithm of negatively associated random variables. In the one-indexed random sequence case, such inequalities for negatively associated random variables are established by Shao (2000) by using his comparison theorem between negatively associated and independent random variables. In the multi-indexed random field case, the comparison theorem fails. The purpose of this paper is to establish the Kolmogorov exponential inequality as well a moment inequality of the maximum partial sums of a negatively associated random field via a different method. By using these inequalities, the sufficient and necessary condition for the law of the iterated logarithm of a negatively associated random field to hold is obtained.