In this paper it is established that the normalized sample paths of a $\phi$-mixing sequence converge to the Brownian motion, under the Lindeberg's condition and under some stationarity assumptions. No mixing rate is required.
"An Invariance Principle for $\phi$-Mixing Sequences." Ann. Probab. 13 (4) 1304 - 1313, November, 1985. https://doi.org/10.1214/aop/1176992814