Empirical Quantile CLTs for Time Dependent Data

Abstract: We establish empirical quantile process CLTs based on $n$ independent copies of a stochastic process ${X_t: t \in E}$ that are uniform in $t \in E$ and quantile levels $\alpha \in I$, where $I$ is a closed sub-interval of $(0,1)$. Typically $E=[0,T]$, or a finite product of such intervals. Also included are CLT's for the empirical process based on ${I_{X_t \le y} - \rm {Pr}(X_t \le y): t \in E, y \in R }$ that are uniform in $t \in E, y \in R$. The process ${X_t: t \in E}$ may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.