# Fractional Brownian motion - probability density function of the increments

I'm starting investigating the properties of the fractionally integrated brownian motion, yet I'm not able to figure out what kind of distribution should an increment of a fBM process follow, conditioned on the previous observations...

For example, when studying an ARMA(1,1) model whose variance follows a GARCH(1,1), we know that the distribution of the next innovation $\epsilon_t$, conditioned on the previous observations, will be gaussian, and it will depend obviously on previous innovations and realizations of the process; by the way, I'm still not able to figure out the conditional distribution for fBM, or even ARFIMA, for that matter.

Is there anybody who can help me with this - or even suggest a reference - that covers the distributional properties of these fractionally integrated models?

Thanks a lot! :)

There is a full derivation of the conditional distribution of fBM in Fink et al: "Conditional distributions of processes related to fractional Brownian motion", J. Appl. Probab. Volume 50, Number 1 (2013), 166-183. The particular theorem you need is 3.1.

You can think of the fractional difference operation as a filtering procedure that "removes" the long memory feature of a serie. However the mathematics for the innovations are the same than for any other ARMA models, and their distributions may be Gaussian, student t or whatever distributions.

The ARMA model is given by :

$\Psi(L) X_{t} = \Phi(L) \epsilon_{t}$

and the ARFIMA by :

$\Psi(L) (1-L)^d X_{t} = \Phi(L) \epsilon_{t}$

The only difference is that the $(1-L)^{d}$ fractional difference operation is applied on $X_{t}$. The innovations $\epsilon_{t}$ are still IID and this permits the MLE method to be used to infers parameters. The $(1-L)^{d}$ can be seen as an AR filtering with infinite lags. Once the fractional difference is applied on the serie, you are left with a standard ARMA model.

A really good reference, easy to follow, on long memory is the book :

Statistics for Long-Memory Processes by Jean Baran - 1994.