Brownian forgery of statistical dependences

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.

13 pages, 2 figures, formatting based on revtex4; v2: revised proof of extended forgery and minor changes; v3: additional discussion on practical implementation and minor edits, published version