I want to get a little help from fbm.
fBm $B_H(t)$ have below characteristics.
(1) stationary increments (Gaussian).
(2) Hurst parmeter $H > \frac{1}{2} $, it is positively correlated.
Therefore, $X(t)=B_H(t)-B_H(t-1)$ ($\text{t is integer}$) is discrete time Guassian process with identically distributed but not independent.
Here is the point.
When getting the sample $x_1, x_2, … , x_n$ … from $X(1), X(2), … X(n)…$ I make the histogram of these sample.
It looks Gaussian. How can it happen?
First, I think it would be not Gaussian due to its dependence (correlation). But in reality, it is Gaussian.
I think it is not due to CLT. it is not about the sum of RVs. Furthermore, if $X(t)$ is not Guassian, its histogram is also not.
Could you give me your little intuition?
Thank you for reading!
Best Answer
It actually is a form of CLT. Even though the $X_n$ are not independent, they are weakly dependent - $X_1$ and $X_n$ are very nearly independent, for large $n$. See weakly dependent CLT for example. At a glance, I found this paper which may interest you.