Feuerverger (1993) and Székely, Rizzo and Bakirov (2007) introduced the notion of distance covariance/correlation as a measure of independence/dependence between two vectors of arbitrary dimension and provided limit theory for the sample versions based on an iid sequence. The main idea is to use characteristic functions to test for independence between vectors, using the standard property that the characteristic function of two independent vectors factorizes. Distance covariance is a weighted version of the squared distance between the joint characteristic function of the vectors and the product of their marginal characteristic functions. Similar ideas have been used in the literature for various purposes: goodnes -of-fit tests, change point detection, testing for independence of variables,... ; see work by Meintanis, Huskova, and many others. In contrast to Székely et al. who use a weight function which is infinite on the axes, the latter authors choose probability density weights. Z. Zhou (2012) extended distance correlation to time series models for testing dependence/independence in a time series at a given lag. He assumed a "physical dependence measure''.

In our work we consider the distance covariance/correlation for general weight measures, finite or infinite on the axesor at the origin. These include the choice of Székely et al., probability and various Lévy measures. The sample versions of distance covariance/correlation are obtained by replacing the characteristic functions by their sample versions. We show consistency under ergodicity and weak convergence to an unfamiliar limit distribution of the scaled auto- and cross-distance covariance/correlation functions under strong mixing. We also study the auto-distance correlation function of the residual process of an autoregressive process. The limit theory is distinct from the corresponding theory of an iid noise process. We illustrate the theory for simulated and real data examples.