Can the bivariate Hurst exponent be higher than an average of the separate Hurst exponents?

TL;DR

This paper examines the theoretical relationships between the bivariate Hurst exponent and the average of individual Hurst exponents, establishing that the bivariate exponent cannot exceed this average, with implications for time series analysis.

Contribution

The paper clarifies the theoretical bounds of the bivariate Hurst exponent, showing it cannot be higher than the average of the separate exponents, and discusses finite sample effects.

Findings

H_{xy} can equal or be less than (H_x + H_y)/2

H_{xy} cannot be greater than (H_x + H_y)/2

Finite sample effects influence Hurst exponent estimation

Abstract

In this note, we investigate possible relationships between the bivariate Hurst exponent $H_{xy}$ and an average of the separate Hurst exponents $\frac{1}{2}(H_x+H_y)$. We show that two cases are well theoretically founded. These are the cases when $H_{xy}=\frac{1}{2}(H_x+H_y)$ and $H_{xy}<\frac{1}{2}(H_x+H_y)$. However, we show that the case of $H_{xy}>\frac{1}{2}(H_x+H_y)$ is not possible regardless of stationarity issues. Further discussion of the implications is provided as well together with a note on the finite sample effect.