For many basic random processes (like [fractional Brownian motion](http://en.wikipedia.org/wiki/Fractional_Brownian_motion)) [Hurst exponent](http://en.wikipedia.org/wiki/Hurst_exponent) "H" and [fractal dimensions](http://en.wikipedia.org/wiki/Fractal_dimension) ( 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](http://en.wikipedia.org/wiki/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.



Relationship between fractal dimension and Hurst exponent

> 
> 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. 
> 
> 
> More generally, local and global behavior are decoupled. Stochastic models of the socalled **Cauchy class** separate fractal dimension and Hurst exponent and allow for any combination of the two parameters, without any linear relation. 
> 
> 
> 


[Stochastic Models That Separate Fractal Dimension and Hurst Effect](http://arxiv.org/abs/physics/0109031), Tilmann Gneiting and Martin Schlather (2001).


