We show that the empirical quantile process from an ARMA$(1, q)$ process which is strongly mixing $\Delta_s$, and is either Gaussian or double exponential, converges to a Gaussian process. This result is used to derive model-free one-step-ahead prediction intervals for such processes. Simulations demonstrate where the asymptotic theory can and cannot be applied to small samples.
"Model-Free One-Step-Ahead Prediction Intervals: Asymptotic Theory and Small Sample Simulations." Ann. Statist. 15 (3) 1064 - 1078, September, 1987. https://doi.org/10.1214/aos/1176350493