Since random walk (martingale) is not applicable in this case, we employ the Hurst Exponent (Hurst, 1951) to test the EMH because it affords a measure for both long-term memory and fractality of a time series, has fewer assumptions about the underlying system, and does not assume a normal distribution.
The fractal dimension is equal to D = 2 - H, where H represents the Hurst exponent.
The values of the Hurst Exponent range between 0 and 1 and the values of the fractal dimension range from 1 to 2.
This edition contains a new chapter on volatility in oil prices and manufacturing activity: an investigation of real options, and another on the Hurst exponent
in energy futures prices.
The existence of memory in the process is based on the value of the Hurst exponent.
The parameter H is called the self-similarity exponent or the Hurst exponent.
The Hurst exponent can then be estimated (Briens et al.
Signals belonging to the same hydrodynamic structure exhibit the same trend, and therefore the same Hurst exponent.
The processes, called fractional Brownian motions (fBm), enable us to do that on the basis of the fitted Hurst exponent H (0 < H < 1) over the time interval of interest .
It is customary to estimate it by means of a special Hurst exponent H  defined by the relation D([tau]) [varies] [[tau].
The Hurst exponent
(H) is the measure of the smoothness of fractal time series and can be related to fractal dimension by H = E + 1 - D, where E is the Euclidean dimension (E = 0 for a point, 1 for a line, 2 for a surface, etc.
We have stated earlier that the Hurst Exponent
provides us with a qualitative benchmark for measuring risk.