Fractional Brownian motion---construction via Hilbert space?

Question

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:

1. Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \mathcal{F}, P)$.

2. Let $\{ \phi_i \}_{i \in \mathbb{Z}}$ be an orthonormal basis for $L^2[0, \infty)$.

3. The map $\phi_i \mapsto Z_i $ extends to an isometry $B$ from $L^2[0, \infty)$ to a subspace of $L^2(\Omega)$.

4. The process defined by $B_t(\omega) = B(1_{[0,t]})(\omega)$ is a version of Brownian motion.

Does this extend in some way to the fractional Brownian motion?