Distance covariance for discretized stochastic processes
Dehling, Harold; Matsui, Muneya; Mikosch, Thomas; Samorodnitsky, Gennady; Tafakori, Laleh
Given an iid sequence of pairs of stochastic processes on the unit interval we construct a measure of independence for the components of the pairs. We define distance covariance and distance correlation based on approximations of the component processes at finitely many discretization points. Assuming that the mesh of the discretization converges to zero as a suitable function of the sample size, we show that the sample distance covariance and correlation converge to limits which are zero if and only if the component processes are independent. To construct a test for independence of the discretized component processes we show consistency of the bootstrap for the corresponding sample distance covariance/correlation.
Herold Dehling's research was partially supported by the DFG through the Collaborative Research Grant SFB 823. Muneya Matsui's research is partly supported by the JSPS Grant-in-Aid for Young Scientists B (16k16023). Thomas Mikosch's research was partly supported by an Alexander von Humboldt Research Award. Gennady Samorodnitsky's research was partially supported by the ARO grant W911NF-12-10385 at Cornell University. Laleh Tafakori would like to thank the Australian Research Council for support through Laureate Fellowship FL130100039.
distance covariance; empirical characteristic function; test for independence; stochastic processes