The rate of convergence in the central limit theorem for non-stationary dependent random vectors

Let (Xj, j >= 1) be a strictly stationary sequence of uniformly mixing random variables with zero mean, unit variance and finite fourth moment. Form the vector Sn = [Sigma]j = 1n[alpha]njXj where [alpha]nj = ([alpha]nj1, [alpha]nj2)t, [alpha]nj1, [alpha]nj2 [set membership, variant] R1 and [short parallel][alpha]nj1[short parallel] <= 1, [short parallel][alpha]nj2[short parallel] <= 1. We estimate the rate at which Sn converges to normality. The extension of this result to bounded Rs-valued weights (s >= 1) is immediate.