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In this note, we prove an $L^p$ uniform approximation of the fractionalBrownian motion with Hurst exponent $0 < H < \frac{1}{2}$ by means of a familyof continuous-time random walks imbedded on a given Brownian motion. Theapproximation is constructed via a pathwise representation of the fractionalBrownian 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 rateof convergence of the approximation scheme is $O(\epsilon_k^{p(1-2\lambda)+2(\delta-1)})$ whenever $\max\{0,1-\frac{pH}{2}\}< \delta < 1$, $\lambda \in\big(\frac{1-H}{2}, \frac{1}{2} + \frac{\delta-1}{p}\big)$.

$L^p$ uniform random walk-type approximation for fractional Brownian motion with Hurst exponent $0 < H < \frac{1}{2}$