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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! :)

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2 Answers 2

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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.

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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.

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