A Functional Central Limit Theorem and its Bootstrap Analogue for Locally Stationary Processes with Application to Independence Testing

Under weak moment conditions, we provide a functional central limit theorem (FCLT) for weighted sums of locally stationary processes for two main frameworks which differ in terms of the boundedness of the used function. Since the FCLT itself insinuates beneficial effects of a bootstrap analogue, we transfer our previous results to the bootstrap world using a local block bootstrap approach. Afterwards, we combine real- and bootstrap-world findings to present a testing procedure for independence of locally stationary processes using a weighted distance composed of characteristic functions and its empirical version as a base. In the end, we illustrate the functionality of our test regarding underlying independence as well as dependence of different forms with a simulation study.