I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of all finite dimensional distributios, does this implied the left hand side and the right hand side of the equality have the same distribution?
Furthermore, I wonder if we can estimate H by the total variation approach? For example: if $X(t)$ is Brownian motion and we define $a=t_{i}-t_{i-1}$ and assume that interval $[0,1]$ is equally partitioned. We can show that: $$\sqrt{a}\sum_1^n |B(t_i)-B(t_{i-1})|\to \sqrt{\frac{2}{\pi}}$$ This gives us $H=0.5$ (which is correct for case of Brownian motion). Thus, I came up with algorithm to estimate $H$ as follow:
- With different value of $a$, calculate corresponding value of total variation.
- Running regression of $log(total variation)$ against $log(a)$ to find exponent of $a$, which I think is $H$.
I ran 10000 simulations with Brownian motion of length 10000 and the mean of $a$ is 0.50 and $var=0.01$. I wonder if my understanding about Hurst exponent as the exponent of a in this case is correct?
Many thanks in advance.
