# Brownian forgery of statistical dependences

## TL;DR

This paper introduces a novel Brownian motion-based method to forge statistical dependences and proposes a new independence test, offering insights into nonlinear correlation measures from a physical perspective.

## Contribution

It extends Levy's forgery concept to arbitrary dependences and establishes a Brownian independence test, bridging stochastic processes with statistical dependence analysis.

## Key findings

- New Brownian independence test based on fluctuating paths
- Extension of Levy's forgery to arbitrary dependences
- Potential for engineering nonlinear correlation measures

## Abstract

The balance held by Brownian motion between temporal regularity and randomness is embodied in a remarkable way by Levy's forgery of continuous functions. Here we describe how this property can be extended to forge arbitrary dependences between two statistical systems, and then establish a new Brownian independence test based on fluctuating random paths. We also argue that this result allows revisiting the theory of Brownian covariance from a physical perspective and opens the possibility of engineering nonlinear correlation measures from more general functional integrals.