P. Vanouplines
University Library, Free University Brussels,The Hurst exponent H, has a value of about 0.72 for many natural phenomena. On the other hand, statistically independent processes with finite variances, in the absence of long-run statistical dependencies, should give H=0.5. And, as cited by Feder (1988), it is shown that in that case:
Note that pi could be derived from the previous equation by performing a rescaled range analysis on the digits of pi-3 if the digits of pi-3 were perfectly random. This idea does not work at all, as we will see in the next chapter.
The relation between the Hurst exponent and the fractal dimension is simply D=2-H. That means that for statistically independent fractional Brownian movement, with D=1.5, the Hurst exponent should be H=0.5, as said before. For the 'various' natural phenomena with H being about 0.72, the fractal dimension D is about 1.28. Remember from the previous chapter that this corresponds to a 'rather smooth' profile-like curve.
A Hurst exponent of 0.5<H<1 corresponds with a profile-like curve showing persistent behaviour. Persistence means that if the curve has been increasing for a period, it is expected to continue for another period. Recall here the phenomenon of 'persistence of rainfall'. If it has been raining for twenty minutes, a rather reliable forecast in many regions of the world is to say that it will continue raining for another twenty minutes. No wonder that it is proven for many regions on earth that a reliable weather forecasting method is to predict for tomorrow the same weather as we have today (and not only for the Sahara, where it will again not rain tomorrow!).
A Hurst exponent of 0<H<0.5 shows antipersistent behaviour. After a period of decreases, a period of increases tends to show up. The antipersistent behaviour has a rather high fractal dimension (1.5<D<2), corresponding to a very 'noisy' profile-like curve (which highly fills up the plane).
Now that we understand the rescaled range analysis, and know what the meaning of a particular value of the Hurst exponent involves, we are ready to move to the analysis of the digits of pi-3.
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Last updated on Sunday 19 November 1995.
©Patrick Vanouplines.