Upper bounding the distance covariance of bounded random vectors

TL;DR

This paper derives an upper bound for the distance covariance of bounded random vectors, linking it to their dimensionality and bounds, with simplified cases for vectors with equal components or single components.

Contribution

It introduces a classical inequality-based method to bound the distance covariance of bounded vectors, providing simplified bounds for specific cases.

Findings

Distance covariance is bounded by a function of dimension and bounds.

Simplified bounds are derived for vectors with equal components.

Special case bounds are provided for single-component vectors.

Abstract

A classical statistical inequality is used to show that the distance covariance of two bounded random vectors is bounded from above by a simple function of the dimensionality and the bounds of the random vectors. Two special cases that further simplify the result are considered: one in which both random vectors have the same number of components, each component taking values in an interval of unit length, and the other in which both random vectors have one component.