Bahadur has obtained an asymptotic almost sure representation for empirical quantiles of independent and identically distributed random variables. In this paper we present an analogous result for a large class of stationary linear processes.
"A Bahadur-Type Representation for Empirical Quantiles of a Large Class of Stationary, Possibly Infinite-Variance, Linear Processes." Ann. Statist. 18 (3) 1188 - 1202, September, 1990. https://doi.org/10.1214/aos/1176347746