We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Electron. J. Probab. 11 (2006), no. 18, 434--459.
"On the small deviation problem for some iterated processes." Electron. J. Probab. 14 1992 - 2010, 2009. https://doi.org/10.1214/EJP.v14-689