Rate of Convergence in the SLLN for Associated Sequences and Fields

A Note on the Convergence Rate of Strong Law of Large Numbers for Positively Associated Sequences

“…The main purpose of this article is to investigate the convergence rate of strong law of large numbers for positively associated random variables in terms of some inequalities obtained. The results obtained extend and improve the relevant ones in Vronskii (1999).…”

“…Theorem 2.2 extends Theorem 5 in Vronskii (1999) from associated case to positively associated setting. In addition, by Theorem 2.2, the requirement of the moment of certain order for the positively associated random variables is given from the condition (2.4), i.e., the result of Theorem 2.1 (2.3) describing the convergence rate of strong law of large numbers.…”

“…Theorem 2.1 generalizes Theorem 3 in Vronskii (1999) from associated sequences to positively associated sequences and improves the latter. In fact, the latter used the condition n=1 u n / k−2+ n k−2 < for an even k > 2 v − 2 / 2 − 1 to get (2.3).…”

“…Hence, the in (2.1) can take values in the open interval 0 1 when we take suitable values for 1 and 2 , i.e., the condition (2.1) can become u n ≤ Cn − for ∈ 0 1 as 1 and 2 are suitably given. So, the requirement on the covariance function u n in Theorem 2.1 is weaker than the corresponding one on the corresponding covariance function u n in Theorem 3 of Vronskii (1999).…”

“…It is well known that associated sequences play an important role in a wide variety of areas, including multivariate statistical analysis, reliability theory and life sciences. One can refer to Newman (1980); Newman and Wright (1982); Newman (1984Newman ( , 1990, Cox and Grimmett (1984), Birkel (1988aBirkel ( ,b, 1989, Roussas (1991Roussas ( , 1993Roussas ( , 1994Roussas ( , 1995Roussas ( , 1996, Yang (2001); Yang et al (2008), and Vronskii (1999) for further comprehension. The concept of association, however, is strictly stronger than the one of positive association in Definition 1.1, as mentioned in Xing and Yang (2010).…”