Asymptotics for random functions moderated by dependent noise

We study limit theorems for a wide class of multi-parameter stochastic processes which are driven by a noise process which may be weakly or even long-range dependent. The processes under study arise in diverse areas and fields such as functional data analysis, life science, engineering and finance. It turns out that under fairly weak conditions on the underlying noise process the limiting law of the corresponding partial sum process is a consequence of the weak convergence of the sequential empirical Kiefer process. The asymptotic limit theory covers the classical large sample situation as well as a general change-point model which extends the location-scale model often considered in change-point analysis. The scope of the results is illustrated by various applications.