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

For a sequence of dependent square-integrable random variables and a sequence of positive constants {bn,n>=1}, conditions are provided under which the series converges almost surely as n-->[infinity] and {Xn,n>=1} obeys the strong law of large numbers almost surely. The hypotheses stipulate that two series converge, where the convergence of the first series involves the growth rates of and {bn,n>=1} and the convergence of the second series involves the growth rate of .