Approximation of rough paths of fractional Brownian motion

We consider a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in ]1\mathrm{/}4,1\mathrm{/}2[$. We give an approximation result in a modulus type distance, up to the second order, by means of a sequence of rough paths lying above elements of the reproducing kernel Hilbert space.