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 (b(n), n >= 11, conditions are provided under which the series Sigma(n)(i=1) (X-i - EXi)/b(i) converges almost surely as it n -> infinity and {X-n, n >= 1} obeys the strong law of large numbers lim(n ->infinity) Sigma(n)(i=1) (X-i -EXi)/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) vertical bar COv (X-n, Xn+k)vertical bar, k >= 1). (C) 2008 Elsevier B.V. All rights reserved.