I am interested in fractional Brownian motion recently.
But I have a question about its Long range dependence(LRD)
$\text{For fractional Brownian motion}$ $B(t), X(t)=B(n+1)-B(n)$ $\text{, n is integer, has long range dependence. }$
$\text{it means its auto-correlation }$ $ \lim_{|i-j|\to\infty}\rho_{i,j} = 0 $
$\text{In this case, we can say }$ $X(i)$ $\text{ and }$ $X(j)$ $\text{ are uncorrelated when }$ $|i-j|\to\infty$
$But \text{ could we say } $$X(i)$ $\text{ and }$ $X(j)$ $\text{are nearly independent in this limit case?}$
I think we couldn't because correlation is different from independence.
Please give me a little help.
Thank you for reading.
Best Answer
I finally got an answer!
We can say they are nearly independent in this limit case, due to its normality.
$X(t)$$\text{ follows Gaussian distribution because it is from fBm}$
$\text{For Gaussian distributions, zero covariance implies independence.}$