(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.)

It seems strange but, even after consulting several books, and hours spent on google, nothing came out about a law of iterated logarithm for the fractional Brownian motion.

I just need a precise reference, on where I can find such a law.

EDIT: My goal is to prove that the fractional Brownian motion of hurst parameter $0<H<1$ has not $H$-Holder continuous trajectories; for the standard BM this can be done in a few lines by exploiting the "standard LIL"; thus, I thought in the fractional case, this can be done in a similar way.

EDIT 2: The law I'm searching for was proved in the 70's, when the expression "fractional Brownian motion" wasn't in use yet; this a reason for which I can't find so much material on the web. Maybe "gaussian process" is better than "fractional BM", as a research advice.

EDIT 3: In the book "Probability and Its Applications" by J. Gani, C.C. Heyde, P. Jagers, T.G. Kurtz, at page 11 holder regularity of fBM is discussed; it's proved that a.s. the $H$-fBM has $\alpha$-Holder continuous trajectories for all $0<\alpha<H$; then when they shows the trajectories are not $H$-Holder, they exploits the following limit $$ \limsup_{t\to0+}\frac{|B_t^H|}{t^H\sqrt{\log\log(1/t)}}=c_H $$ where $c_H$ is a "suitable constant"; thus we know that this $\limsup$ is finite and clearly $\ge0$, but can $c_H$ be zero? The reference the authors give is this paper by M. Arcones; I'm looking in it, but till now I didn't found nothing "clean" as the $\limsup$ above.

  • $\begingroup$ 1. Such way of cross-posting is against SE policy, I believe. 2. My answer on math.SE (the first point) directly addresses your "goal". $\endgroup$ – zhoraster Sep 18 '17 at 6:09
  • $\begingroup$ I don't think this post is disrespectful neither to you nor the SE community. You provided an helpful answer. But here I am asking for a precise reference on the LIL for the fractional BM, which is not given on MSE. That's all. $\endgroup$ – Joe Sep 18 '17 at 8:53
  • $\begingroup$ In many respects it is disrespectful. Firstly, you didn't disclose on math.SE the non-Holder "goal" you write here. And my answer (point 1) given almost two weeks ago does directly address this question: the exact modulus of continuity I wrote means that fBm in not $H$-Holder continuous. Secondly, after asking for a precise reference you didn't wait long enough for an answer, which is already there. Thirdly, you didn't inform neither this nor that communities about the cross-posting. I believe that all these three points abuse SE norms. $\endgroup$ – zhoraster Sep 18 '17 at 9:33
  • $\begingroup$ Probably you are right. I ask sorry, sincerely. I'm doing this, simply because I have to conclude my master degree thesis, I do not have so much time and sometimes the panic wins on me. $\endgroup$ – Joe Sep 18 '17 at 9:56

This has been proved by S. Orey in this article:

Steven Orey. Growth rate of certain Gaussian processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II : Probability theory. Berkeley, Calif. : Univ. California Press, 443-451, 1972.

Note that the growth rate is equivalent to the behavior at 0 by time inversion. You can get as well the LIL at any point by increments stationarity.


There is formula here (item 3):


and how about that:

Séminaire de Probabilités XLI pp 161-179 Part of the Lecture Notes in Mathematics book series (LNM, volume 1934) A Law of the Iterated Logarithm for Fractional Brownian Motions

Driss BarakaThomas Mountford

  • $\begingroup$ many thanks. However, in the post of MSE the author is not sure about the form of LIL he wrote; nevertheless I didn't found anything in the references he gave. Next, I know the article by Baraka-Mountford, but there the authors talk about local times and no clean formula like the one suggested in MSE post is given. I am searching for that kind of formula, with a reference. $\endgroup$ – Joe Sep 17 '17 at 10:32

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