An Invariance Principle for Certain Dependent Sequences

Let $X_1, X_2, \cdots$ be a strictly stationary second order sequence which is "associated"; i.e., is such that any two coordinatewise nondecreasing functions of the $X_i$'s (of finite variance) are nonnegatively correlated. If $\sum_j \operatorname{Cov}(X_1, X_j) < \infty$, then the partial sum processes, $W_n(t)$, defined in the usual way so that $W_n(m/n) = (X_1 + \cdots + X_m - mE(X_1))/\sqrt n$ for $m = 1, 2, \cdots$, converge in distribution on $C\lbrack 0, T\rbrack$ 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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