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}{bn,n≥1}, conditions are provided under which the series ∑i=1n(Xi−EXi)/bi converges almost surely as n→∞n→∞ and {Xn,n≥1}{Xn,n≥1} obeys the strong law of large numbers limn→∞∑i=1n(Xi−EXi)/bn=0 almost surely. The hypotheses stipulate that two series converge, where the convergence of the first series involves the growth rates of {VarXn,n≥1} and {bn,n≥1}{bn,n≥1} and the convergence of the second series involves the growth rate of {supn≥1|Cov(Xn,Xn+k)|,k≥1}.