On the Weak Convergence of the Function-Indexed Sequential Empirical Process and its Smoothed Analogue under Nonstationarity

We study the sequential empirical process indexed by general function classes and its smoothed set-indexed analogue. Sufficient conditions for asymptotic equicontinuity are provided for nonstationary arrays of time series. This yields comprehensive general results that are applicable to various notions of dependence, which is exemplified in detail for nonstationary $\alpha$-mixing series. Especially, we obtain the weak convergence of the sequential process under essentially the same mild assumptions as known for the classical empirical process. Core ingredients of the proofs are a novel maximal inequality for nonmeasurable stochastic processes, uniform chaining arguments and, for the set-indexed smoothed process, uniform Lipschitz properties.