  In this note, we prove an $L^p$ uniform approximation of the fractional
Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a family
of continuous-time random walks imbedded on a given Brownian motion. The
approximation is constructed via a pathwise representation of the fractional
Brownian motion in terms of a standard Brownian motion. For an arbitrary choice
$\epsilon_k$ for the size of the jumps of the family of random walks, the rate
of convergence of the approximation scheme is $O(\epsilon_k^{p(1-2\lambda)+
2(\delta-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}< \delta < 1$, $\lambda \in
\big(\frac{1-H}{2}, \frac{1}{2} + \frac{\delta-1}{p}\big)$.
