Empirical Quantile CLTs for Time Dependent Data

We establish empirical quantile process CLTs based on n independent copies of a stochastic process {Xt : t ∈ E} that are uniform in t ∈ E and quantile levels α ∈ I, where I is a closed subinterval 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 {IXt≤y − Pr(Xt ≤ y) : t ∈ E, y ∈ R} that are uniform in t ∈ E, y ∈ R. The process {Xt : t ∈ E} may be chosen from a broad collection of Gaussian processes, compound Poisson processes, stationary independent increment stable processes, and martingales.