Distance covariance for discretized stochastic processes

TL;DR

This paper introduces a method to measure independence between discretized stochastic processes using distance covariance and correlation, proving consistency and convergence properties for the measures and bootstrap-based tests.

Contribution

It develops a new approach to assess independence of stochastic processes through discretization-based distance covariance, with proven convergence and bootstrap test validity.

Findings

Sample distance covariance converges to zero if and only if processes are independent.

Bootstrap method is consistent for testing independence.

Method effectively captures dependence structure in discretized processes.

Abstract

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.