In an earlier paper, the author proves an invariance principle for mixingales, a generalization of the concepts of mixing sequences and martingale differences, under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. In this note we relax further the required degree of stationarity, requiring only that the squared variables properly normalized form a uniformly integrable family, and the partial sums have variances consistent with the Wiener process.
"On the Invariance Principle for Nonstationary Mixingales." Ann. Probab. 5 (4) 616 - 621, August, 1977. https://doi.org/10.1214/aop/1176995772