  We consider a special class of weak dependent random variables with control
on covariances of Lipschitz transformations. This class includes, but is not
limited to, positively, negatively associated variables and a few other classes
of weakly dependent structures. We prove a Strong Law of Large Numbers with a
characterization of convergence rates, which is almost optimal, in the sense
that it is arbitrarily close to the optimal rate for independent variables.
Moreover, we prove an inequality comparing the joint distributions with the
product distributions of the margins, similar to the well-known Newman's
inequality for characteristic functions of associated variables. As a
consequence, we prove a Central Limit Theorem together with its functional
counterpart, and also the convergence of the empirical process for this class
of weak dependent variables.
