Conditional Distributions and Innovations in Fractional Models
Summary
The document asks how to characterize an increment of fractional Brownian motion conditional on past observations, and how that question relates to ARFIMA models. It points to a published derivation of conditional distributions for processes related to fractional Brownian motion, specifically identifying a theorem as a relevant reference. This gives readers a route to the detailed probability result, though the document itself does not reproduce its assumptions or derivation.
For ARFIMA, the response describes fractional differencing as an infinite-lag filter that removes long-memory behavior and leaves an ARMA representation in the transformed series. The innovations remain independent and identically distributed under the stated model setup, with their distribution chosen according to the modeling assumptions, such as Gaussian or Student-t. That framing supports likelihood-based parameter estimation. The distinction matters: this innovation claim concerns the ARFIMA representation and should not be mistaken for a full conditional-distribution formula for fractional Brownian motion. The short discussion offers a reference but does not address model-specific estimation conditions or distributional alternatives in depth.
Key ideas
- Fractional Brownian motion has a conditional-distribution derivation available in the cited research literature.
- Fractional differencing can be viewed as an infinite-lag filter for long-memory series.
- An ARFIMA series becomes an ARMA-type representation after fractional differencing.
- The innovations in that representation are modeled as independent and identically distributed, with the distribution specified by the analyst.
- The ARFIMA explanation does not itself provide the conditional increment law for fractional Brownian motion.
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# Fractional Brownian motion - probability density function of the increments
# 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! :)
## Answer by Forgottenscience (score 2)
https://quant.stackexchange.com/a/32587
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.
## Answer by Malick (score 2)
https://quant.stackexchange.com/a/32604
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.