The most striking conclusion of this paper is that measuring the Hurst parameter, even in artificial data, is very hit and miss. (realclimate.org)
Figure 3: (a) Southern Oscillation Index (SOI) data (black) versus artificial data proportional to the SOI, and with normally - distributed white noise and a sinusoidal signal added (red). (skepticalscience.com)
Can you show us the effect Zach refers to with some artificial data (decreasing noise amplitude with time)? (tamino.wordpress.com)