Weak convergence of sequential kernel smoothed empirical process of the residuals in AR(p) models

This paper first extends the weak convergence result of the sequential empirical processes to the sequential kernel smoothed empirical process. Then we consider the residual-based sequential kernel smoothed empirical process. Under some mild conditions, the sequential kernel smoothed empirical process of the residuals in AR(p) model is also proved to converge weakly to the Kiefer process. Thus this paper generalizes the classical convergence result about the sequential empirical processes to the kernel smoothed sequential process, and the kernel smoothed sequential process of the residuals in AR(p)-model, thereby enhancing the smoothness of the process while preserving the convergence. Furthermore, the results can provide theoretical justification for using smoothed residual-based statistics in change-point detection.