A note on the strong law of large numbers for associated sequences (Q2368440)

Summary: We prove that the sequence \(\{b_n^{-1}\sum^n_{i=1}(X_i-EX_i)\}_{n\geq 1}\) converges a.e.~ to zero if \(\{X_n,n\geq 1\}\) is an associated sequence of random variables with \(\sum^\infty_{n=1} b_{k_n}^{-2}\text{Var}(\sum^{k_n}_{i=k_n+1}X_i) <\infty\) where \(\{b_n,n\geq 1\}\) is a positive nondecreasing sequence and \(\{k_n, n\geq 1\}\) is a strictly increasing sequence, both tending to infinity as \(n\) tends to infinity and \(0<a=\inf_{n\geq 1}b_{k_n}b_{k_{n+1}}^{-1}\leq \sup_{n\geq 1} b_{k_n}b_{k_{n+1}}^{-1}=c<1\).