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 and weak convergence are provided for nonstationary arrays of time series, in terms of uniform moment bounds for partial sums and, for the set-indexed smoothed process, $L_p$-Lipschitz regularity. This yields comprehensive general results on the weak convergence of sequential empirical processes, which are applicable to various notions of dependence. Especially, we show that our moment conditions imply the weak convergence of the sequential process under essentially the same mild assumptions (on the degree of dependence and the complexity of the indexing function class) as known for the classical empirical process. This is exemplified in detail for nonstationary $\alpha$-mixing time series. Core ingredients of the proofs are a novel maximal inequality for nonmeasurable stochastic processes, uniform chaining arguments and suitable pathwise uniform Lipschitz properties.