On convergence properties of sums of dependent random variables under second moment and covariance restrictions

Abstract For a sequence of dependent square-integrable random variables and a sequence of positive constants { b n , n ≥ 1 } , conditions are provided under which the series ∑ i = 1 n ( X i − E X i ) / b i converges almost surely as n → ∞ and { X n , n ≥ 1 } obeys the strong law of large numbers lim n → ∞ ∑ i = 1 n ( X i − E X i ) / b n = 0 almost surely. The hypotheses stipulate that two series converge, where the convergence of the first series involves the growth rates of { Var X n , n ≥ 1 } and { b n , n ≥ 1 } and the convergence of the second series involves the growth rate of { sup n ≥ 1 | Cov ( X n , X n + k ) | , k ≥ 1 } .