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

AbstractLet (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 = Σj = 1nαnjXj where αnj = (αnj1, αnj2)t, αnj1, αnj2 ∈ R1 and ∥αnj1∥ ≤ 1, ∥αnj2∥ ≤ 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