Convergence rates for the strong law of large numbers under association

Abstract

We prove convergence rates for the Strong Laws of Large Numbers (SLLN) for associated variables which are arbitrarily close to the optimal rates for independent variables. A rst approach is based on exponential inequalities, a usual tool for this kind of problems. Following the optimization e orts of several authors, we improve the rates derived from exponential inequalities to log2 n n1=2 . A more recent approach tries to use maximal inequalities together with moment inequalities. We prove a new maximal order inequality of order 4 for associated variables, using a telescoping argument. This inequality is then used to prove a SLLN convergence rate arbitrarily close to log1=4 n n1=2 .