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

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.