(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](http://people.math.binghamton.edu/arcones/prep/gl.pdf); I'm looking in it, but till now I didn't found nothing "clean" as the $\limsup$ above.



Reference for LIL for fractional Brownian motion
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.


