This paper introduces a new mixing condition for stationary processes which is weaker than $\phi$-mixing but stronger than strong mixing. Many processes arising in applications, e.g., first order autoregressive processes, obey the conditions. The main result is that the empiric cdf of a sample from such processes converges to a Gaussian process.
"The Asymptotic Distribution Theory of the Empiric CDF for Mixing Stochastic Processes." Ann. Statist. 3 (4) 809 - 824, July, 1975. https://doi.org/10.1214/aos/1176343184