Let X₁, X₂, ⋯ be a strictly stationary second order sequence which is "associated"; i.e., is such that any two coordinatewise nondecreasing functions of the Xᵢ's (of finite variance) are nonnegatively correlated. If ∑ⱼ Cov(X₁, Xⱼ) < ∞, then the partial sum processes, Wₙ(t), defined in the usual way so that Wₙ(m/n) = (X₁ + ⋯ + Xₘ - mE(X₁))/√ n for m = 1, 2, ⋯, converge in distribution on C 0, T to a Wiener process. This result is based on two general theorems concerning associated random variables which are of independent interest.
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Newman et al. (1981) studied this question.