Empirical process theory for locally stationary processes

We provide a framework for empirical process theory of locally stationary processes using the functional dependence measure. Our results extend known results for stationary mixing sequences by another common possibility to measure dependence and allow for additional time dependence. We develop maximal inequalities for expectations and provide functional limit theorems and Bernstein-type inequalities. We show their applicability to a variety of situations, for instance we prove the weak functional convergence of the empirical distribution function and uniform convergence rates for kernel density and regression estimation if the observations are locally stationary processes.