Relationship between fractal dimension and Hurst exponent

For many basic random processes (like fractional Brownian motion) Hurst exponent "H" and fractal dimensions ( Hausdorf = Minkoswki typically) "D" are realated by simple formula D = 2-H. I want to clarify what are exact statemetns known about this relationships.

Consider some function y = f(t) (not a random process, but just function). Assume that fractal dimensions of its graph equals to "D".

Is it true that Hurst exponent is somehow correctly defined and equals to H=2-D ?

There possibly can be some subtleties defining the Hurst exponent. Let me try to give the following precise definition. Assume function f(t) is defined on the interval [0, 1]. Let us look on the values of function at points t=k/n, k=0...n. So we obtain discrete time series f_0, ...,f_n. And use the standard way of calculating the rescaled range $(R/S)_{n}$, and existence of the Hurst exponent means that $(R/S)_{n}$ ~ $C n^H$. I am not quite sure this definition is the correct one, but I think it can be suitable modified.

• Do you want your function to be continuous? – Goldstern Oct 19 '13 at 23:19
• @Goldstern Yes, I mean continuous. – Alexander Chervov Oct 20 '13 at 16:10

In principle, fractal dimension and Hurst exponent are independent of each other: fractal dimension is a local property, while the long-memory dependence characterized by the Hurst exponent is a global characteristic. For self-affine processes, the local properties are reflected in the global ones, resulting in the celebrated relationship $D+H=n+1$ between fractal dimension $D$ and Hurst exponent $H$ for a self-affine surface in $n$-dimensional space.