Empirical process theory for locally stationary processes. (English) Zbl 1486.60064
Functional dependence measure, which uses a Bernoulli shift representation and decomposition into martingales and \(m\)-dependent sequences, became popular recently as an alternative measure of dependence among random variables but no general empirical process theory (allowing for general classes of functions) using this measure is available. In this paper, maximal inequalities and functional central limit theorems of empirical processes are proved under functional dependence. The connections and comparisons of these results to that under other measures of dependences, such as that defined by Markov chains and mixing coefficients, are also studied. This paper works in the framework of locally stationary processes and therefore automatically provides the first general empirical process theory in this setting. As an application, uniform convergence rates for nonparametric regression and M-estimators with locally stationary noise are obtained.
Reviewer: Dongsheng Tu (Kingston)
Keywords:
empirical process theory; functional central limit theorem; functional dependence measure; locally stationary processes; maximal inequalityReferences:
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