When I look at the code for the
compute_Hc function in the Hurst package for Python, there is an initial finite differencing step. Everything else after that agrees with Wikipedia's description of the Hurst exponent except it works with the derivative of the series, instead of the original series.
random_walk function, which is supposed to use Fractional Brownian Motion, seems to be in agreement with their
compute_Hc function. If you remove the differencing step, then there ends up being disagreement. But then is their
random_walk function correctly implemented? Because if it is correctly implemented, then the Wikipedia article needs to be corrected.