  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.
