A general method to the strong law of large numbers and its applications

Abstract

A general method to prove the strong law of large numbers is given by using the maximal tail probability. As a result the convergence rate of S n / n for both positively associated sequences and negatively associated sequences is n - 1 / 2 ( log n ) 1 / 2 ( log log n ) δ / 2 for any δ > 1 . This result closes to the optimal achievable convergence rate under independent random variables, and improves the rates n - 1 / 3 ( log n ) 2 / 3 and n - 1 / 3 ( log n ) 5 / 3 obtained by Ioannides and Roussas [1999. Exponential inequality for associated random variables. Statist. Probab. Lett. 42, 423–431] and Oliveira [2005. An exponential inequality for associated variables. Statist. Probab. Lett. 73, 189–197], respectively. In this sense the proposed general method may be more effective than its peers provided by Fazekas and Klesov [2001. A general approach to the strong law of large numbers. Theory Probab. Appl. 45(3), 436–449] and Ioannides and Roussas [1999. Exponential inequality for associated random variables. Statist. Probab. Lett. 42, 423–431].